CHRONOMETRIC INVARIANTSzelmanov.ptep-online.com/books/zelmanov1944.pdf · 2010. 3. 13. · 300 N. Zeeb Road, P.O. Box 1346, Ann Arbor, MI 48106-1346, USA ... Prof. Florentin Smarandache,

ABRAHAM ZELMANOV

CHRONOMETRIC

INVARIANTS

ON DEFORMATIONS AND THE CURVATURE OFACCOMPANYING SPACE

AMERICANRESEARCH

PRESS

Abraham Zelmanov

CHRONOMETRICINVARIANTS

On Deformations and the Curvature ofACCOMPANYING SPACE

(1944)

Translated from the Russianand edited by

Dmitri Rabounski and Stephen J. Crothers

Formulae correction byLarissa Borissova

2006American Research Press

Rehoboth, New Mexico, USA

Originally written in Russian by Abraham Zelmanov, 1944. Translated into Englishand edited by Dmitri Rabounski and Stephen J. Crothers, 2006. Please send all com-ments on the book to the Editors: [email protected]; [email protected]

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Contents

Foreword of the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Introducing Chronometric Invariants (by the Editor) . . . . . . . . . . . . 13

Chapter 1

PRELIMINARY NOTICES

§1.1 The initial suppositions of today’s relativistic cosmology 17

§1.2 The world-metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

§1.3 Properties of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

§1.4 The law of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

§1.5 The law of gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

§1.6 The system of the cosmological equations. . . . . . . . . . . . . . . .23

§1.7 Its solutions. The main peculiarities . . . . . . . . . . . . . . . . . . . . . 24

§1.8 Kinds of non-empty universes . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

§1.9 Sections of relativistic cosmology . . . . . . . . . . . . . . . . . . . . . . . . 28

§1.10 Accompanying space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

§1.11 The cosmological equations: the regular interpretation . . 32

§1.12 Numerous drawbacks of homogeneous models . . . . . . . . . . . 34

§1.13 Some peculiarities of the evolution of the models . . . . . . . . 36

§1.14 The homogeneous models and observational data . . . . . . . . 38

§1.15 Non-uniformity of the visible part of the Universe . . . . . . 42

§1.16 The theory of an inhomogeneous universe . . . . . . . . . . . . . . . 44

§1.17 The cosmological equations: the local interpretation . . . . . 47

§1.18 Friedmann’s case of an inhomogeneous universe . . . . . . . . 50

§1.19 The mathematical methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

§1.20 Relativistic physical equations. . . . . . . . . . . . . . . . . . . . . . . . . . . .55

§1.21 Numerous cosmological consequences. . . . . . . . . . . . . . . . . . . .58

4 Contents

Chapter 2

THE MATHEMATICAL METHODS

§2.1 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

§2.2 An observer’s reference space. Sub-tensors. . . . . . . . . . . . . .64

§2.3 Time, co-quantities, and chr.inv.-quantities . . . . . . . . . . . . . . 66

§2.4 The potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

§2.5 Chr.inv.-differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

§2.6 Changing the order of chr.inv.-differentiation. . . . . . . . . . . .71

§2.7 The power quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

§2.8 The space metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

§2.9 The chr.inv.-vector of velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

§2.10 The chr.inv.-tensor of the rate of space deformations. . . .82

§2.11 Deformations of a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

§2.12 Transformations of space elements. . . . . . . . . . . . . . . . . . . . . . .88

§2.13 Christoffel’s symbols in chr.inv.-form. . . . . . . . . . . . . . . . . . . .89

§2.14 General covariant differentiation in chr.inv.-form . . . . . . . 91

§2.15 The Riemann-Christoffel tensor in chr.inv.-form . . . . . . . . 93

§2.16 Chr.inv.-rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

§2.17 The chr.inv.-vector of the angular velocities of rotation.Its chr.inv-rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

§2.18 Differential rotations. Differential deformations . . . . . . . . 101

§2.19 Differential rotations of a space . . . . . . . . . . . . . . . . . . . . . . . . . 102

§2.20 The system of locally independent quantities . . . . . . . . . . . 104

§2.21 The chr.inv.-tensors of the space curvature . . . . . . . . . . . . 111

§2.22 On the curvature of space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116

Chapter 3

RELATIVISTIC PHYSICAL EQUATIONS

§3.1 The metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

§3.2 Christoffel’s symbols of the 1st kind . . . . . . . . . . . . . . . . . . . . 124

§3.3 Christoffel’s symbols of the 2nd kind . . . . . . . . . . . . . . . . . . . 127

§3.4 The speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

§3.5 A point-body. Its mass, energy, and the momentum . . . 131

Contents 5

§3.6 Transformations of the energy and momentum of apoint-body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132

§3.7 The time equation of geodesic lines . . . . . . . . . . . . . . . . . . . . . 135

§3.8 The spatial equations of geodesic lines . . . . . . . . . . . . . . . . . . 139

§3.9 The mechanical sense of the power quantities . . . . . . . . . . 141

§3.10 The energy-momentum tensor. . . . . . . . . . . . . . . . . . . . . . . . . .143

§3.11 The time equation of the law of energy . . . . . . . . . . . . . . . . . 147

§3.12 The spatial equations of the law of energy . . . . . . . . . . . . . . 149

§3.13 A space element. Its energy and the momentum . . . . . . . 152

§3.14 Einstein’s covariant tensor. Its time component . . . . . . . . 154

§3.15 Einstein’s covariant tensor. The mixed components . . . . 156

§3.16 Einstein’s covariant tensor. The spatial components . . . . 159

§3.17 The Einstein tensor and chr.inv.-quantities . . . . . . . . . . . . . 170

§3.18 The law of gravitation. Its time covariant equation . . . . . 174

§3.19 The law of gravitation. The mixed covariant equations 174

§3.20 The law of gravitation. The spatial covariant equations 175

§3.21 The scalar equation of gravitation. . . . . . . . . . . . . . . . . . . . . . .176

§3.22 The first chr.inv.-tensor form of the equations of gravi-tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

§3.23 The second chr.inv.-tensor form of the equations ofgravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

§3.24 The structure of the equations of gravitation . . . . . . . . . . . 181

Chapter 4

NUMEROUS COSMOLOGICAL CONSEQUENCES

§4.1 Locally-accompanying frames of reference . . . . . . . . . . . . . 183

§4.2 Accompanying frames of reference . . . . . . . . . . . . . . . . . . . . . 183

§4.3 The kinds of matter. Their motions . . . . . . . . . . . . . . . . . . . . . 185

§4.4 Matter in the accompanying frame of reference . . . . . . . . 186

§4.5 The cosmological equations of gravitation. . . . . . . . . . . . . . .188

§4.6 The cosmological equations of energy. . . . . . . . . . . . . . . . . . .190

§4.7 The main form of the cosmological equations . . . . . . . . . . . 190

§4.8 Free fall. The primary coordinate of time . . . . . . . . . . . . . . 192

6 Contents

§4.9 Characteristics of anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

§4.10 Absolute rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

§4.11 Mechanical anisotropy and geometrical anisotropy . . . . . 199

§4.12 Isotropy and homogeneity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .200

§4.13 Stationarity in a finite volume . . . . . . . . . . . . . . . . . . . . . . . . . . 202

§4.14 Transformations of the mean curvature of space . . . . . . . 203

§4.15 When the mass of an element and its energy remainunchanged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204

§4.16 Criteria of the extreme volumes . . . . . . . . . . . . . . . . . . . . . . . . 206

§4.17 The evolution of the volume of an element . . . . . . . . . . . . . 208

§4.18 Changes of the criterion of the curvature . . . . . . . . . . . . . . . 211

§4.19 The states of the infinite density. . . . . . . . . . . . . . . . . . . . . . . .213

§4.20 The ultimate states of the infinite cleanliness . . . . . . . . . . 214

§4.21 The states of unchanged volume and of the minimumvolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

§4.22 The states of the maximum volume. . . . . . . . . . . . . . . . . . . . .216

§4.23 The transformations, limited from above and below. . . .217

§4.24 The area, where deformations of an element are real . . . .218

§4.25 The kinds of monotonic transformations of a volume. . .220

§4.26 The possibility of an arbitrary evolution of an ele-ment of volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222

§4.27 The kinds of evolution of the element. . . . . . . . . . . . . . . . . . .227

§4.28 The roles of absolute rotation and anisotropy. . . . . . . . . . .228

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

♦

Foreword of the Editor

Abraham Zelmanovin the 1940’s

Abraham Leonidovich Zelmanov was born onMay 15, 1913 in Poltava Gubernya of the Rus-sian Empire. His father was a Judaic religiousscientist, a specialist in comments on Torahand Kabbalah. In 1937 Zelmanov completedhis education at the Mechanical MathematicalDepartment of Moscow University. After 1937he was a research-student at the SternbergAstronomical Institute in Moscow, where hepresented his dissertation in 1944. In 1953 hewas arrested for “cosmopolitism” in Stalin’scampaign against Jews. However, as soon asStalin died, Zelmanov was set free, after some

months of imprisonment. For several decades Zelmanov and hisparalyzed parents lived in a room in a flat shared with neighbours.He took everyday care of his parents, so they lived into old age.Only in the 1970’s did he obtain a personal municipal flat. He wasmarried three times. Zelmanov worked on the academic staff of theSternberg Astronomical Institute all his life, until his death on thewinter’s day, 2nd of February, 1987.

He was very thin in physique, like an Indian yogi, rather shorterthan average, and a very delicate man. From his appearance it waspossible to think that his life and thoughts were rather ordinary oruninteresting. However, in acquaintance with him and his scientificdiscussions in friendly company one formed another opinion abouthim. Those were discussions with a great scientist and humanistwho reasoned in a very unorthodox way. Sometimes we thoughtthat we were not speaking with a contemporary scientist of the20th century, but some famous philosopher from Classical Greeceor the Middle Ages. So the themes of those discussions are eternal— the interior of the Universe, the place of a human being in theUniverse, the nature of space and time.

Zelmanov liked to remark that he preferred to make mathemat-ical “instruments” than to use them in practice. Perhaps thereby

8 Foreword

his main goal in science was the mathematical apparatus of physicalobservable quantities in the General Theory of Relativity knownas the theory of chronometric invariants [1]. In developing theapparatus he also created other mathematical methods, namely— kinemetric invariants [5] and monad formalism [6]. Being verydemanding of himself, Zelmanov published less than a dozen sci-entific publications during his life (see References), so every publi-cation is a concentrate of his fundamental scientific ideas.

Most of his time was spent in scientific work, but he sometimesgave lectures on the General Theory of Relativity and relativisticcosmology as a science for the geometrical structure of the Uni-verse. Stephen Hawking, a young scientist in the 1960’s, attendedZelmanov’s seminars on cosmology at the Sternberg AstronomicalInstitute in Moscow. Zelmanov presented him as a “promisingyoung cosmologist”. Hawking read a brief report at one of thoseseminars.

Because Zelmanov made scientific creation the main goal ofhis life, writing articles was a waste of time to him. However henever regretted time spent on long discussions in friendly company,where he set forth his philosophical concepts on the geometricalstructure of the Universe and the process of human evolution. Inthose discussions he formulated his famous Anthropic Principleand the Infinite Relativity Principle.

His Anthropic Principle is stated here in his own words, in twoversions. The first version sets forth the idea that the law of humanevolution is dependent upon fundamental physical constants:

Humanity exists at the present time and we observe worldconstants completely because the constants bear their specificnumerical values at this time. When the world constants boreother values humanity did not exist. When the constantschange to other values humanity will disappear. That is, hu-manity can exist only with the specific scale of the numericalvalues of the cosmological constants. Humanity is only an epi-sode in the life of the Universe. At the present time cosmo-logical conditions are such that humanity develops.

In the second form he argues that any observer depends on theUniverse he observes in the same way that the Universe dependson him:

The Universe has the interior we observe, because we observethe Universe in this way. It is impossible to divorce the Uni-verse from the observer. The observable Universe depends on

Foreword 9

the observer and the observer depends on the Universe. If thecontemporary physical conditions in the Universe change thenthe observer is changed. And vice versa, if the observer ischanged then he will observe the world in another way, so theUniverse he observes will also change. If no observers existthen the observable Universe as well does not exist.

It is probable that by proceeding from his Anthropic Principle,in the years 1941–1944, Zelmanov solved the well-known problemof physical observable quantities in the General Theory of Rela-tivity.

It should be noted that many researchers were working on thetheory of observable quantities in the 1940’s. For example, Lan-dau and Lifshitz, in their famous The Classical Theory of Fields,introduced observable time and the observable three-dimensionalinterval, similar to those introduced by Zelmanov. But they limitedthemselves only to this particular case and did not arrive at generalmathematical methods to define physical observable quantities inpseudo-Riemannian spaces. It was only Cattaneo, an Italian math-ematician, who developed his own approach to the problem, not farremoved from Zelmanov’s solution. Cattaneo published his resultson the theme in 1958 and later [9–12]. Zelmanov knew those art-icles, and he highly appreciated Cattaneo’s works. Cattaneo alsoknew of Zelmanov’s works, and even cited the theory of chrono-metric invariants in his last publication [12].

In 1944 Zelmanov completed his mathematical apparatus forcalculating physical observable quantities in four-dimensionalpseudo-Riemannian space, in strict solution of that problem. Hecalled the apparatus the theory of chronometric invariants.

Solving Einstein’s equations with this mathematical apparatus,Zelmanov obtained the total system of all cosmological models(scenarios of the Universe’s evolution) which could be possible asderived from the equations. In particular, he had arrived at thepossibility that infinitude may be relative. Later, in the 1950’s, heenunciated the Infinite Relativity Principle:

In homogeneous isotropic cosmological models spatial infinityof the Universe depends on our choice of that reference framefrom which we observe the Universe (the observer’s referenceframe). If the three-dimensional space of the Universe, beingobserved in one reference frame, is infinite, it may be finite inanother reference frame. The same is just as well true for thetime during which the Universe evolves.

10 Foreword

In other words, using purely mathematical methods of the Gen-eral Theory of Relativity, Zelmanov showed that any observerforms his world-picture from a comparison between his observa-tional results and some standards he has in his laboratory — thestandards of different objects and their physical properties. So the“world” we see as a result of our observations depends directlyon that set of physical standards we have, so the “visible world”depends directly on our considerations about some objects andphenomena.

The mathematical apparatus of physical observable quantitiesand those results it gave in relativistic cosmology were the firstresults of Zelmanov’s application of his Anthropic Principle to theGeneral Theory of Relativity. To obtain the results with generalcovariant methods (standard in the General Theory of Relativity),where observation results do not depend on the observer’s refer-ence properties, would be impossible.

Unfortunately, Zelmanov’s scientific methods aren’t very pop-ular with today’s physicists. Most theoreticians working in GeneralRelativity don’t use his very difficult methods of chronometricinvariants, although the methods afford more opportunities thanregular general covariant methods. The reason is that Zelmanovput his scientific ideas into the “code” of this difficult mathematicalterminology. It is of course possible to understand Zelmanov’s ideasusing his mathematical apparatus in detail; he patiently taughtseveral of his pupils. For all other scientists it has proved verydifficult to understand Zelmanov’s mathematical methods from hisvery compressed scientific articles with formulae, without his per-sonal comments.

Herein I present Zelmanov’s dissertation of 1944, where hismathematical apparatus of chronometric invariants has been de-scribed in all the necessary details. The dissertation also containsnumerous results in cosmology which Zelmanov had obtained usingthe mathematical methods. It is impossible to find a more detailedand systematic description of the theory of chronometric invari-ants, than the dissertation. Even the book Elements of the Gener-al Theory of Relativity [8], which Vladimir Agakov had composedfrom Zelmanov’s lectures and articles, gives a very fragmented ac-count of the mathematical methods that prevents a reader fromlearning it on his own. The same can be said about Zelmanov’soriginal papers, each no more than a few pages in length. Anywaythe dissertation is the best for depth of detail. Sometimes Zelmanov

Foreword 11

himself said that to use the mathematical methods of chronometricinvariants in its full power would be possible only after studyinghis dissertation.

The sole surviving manuscript of Zelmanov’s dissertation is keptin the library of the Sternberg Astronomical Institute in Moscow,and the manuscript is in very poor condition. This is the fourthor the fifth typescript with handwritten formulae. From the hand-writing we can conclude that the formulae were inscribed by Zel-manov personally. Some fragments of the manuscript are so fadedthat it is almost impossible to read. I asked Larissa Borissova, whoknew Zelmanov closely, beginning from 1963, to make a copy of themanuscript for me. She did so, and I therefore extend my thanksto her.

In preparation for publication I reconstructed the damaged textfragments in accordance with context. Besides this, I introducednumerous necessary changes to the manuscript, because the termi-nology Zelmanov used in 1944 has became obsolete. For instance,Zelmanov initially called quantities invariant with respect to trans-formation of time “in-invariants”, however in the 1950’s he intro-duced the more useful term “chronometric invariants”. The latterterm has become fixed in the annals of science. Symbols for num-erous tensor quantities have also became obsolete. Therefore I putthe old terms in order in accordance with the contemporary termi-nology of chronometric invariants, which Zelmanov finished in the1960’s.

This book mainly targets an experienced reader, who knowsthe basics of the theory of chronometric invariants and wants tostudy the theory in detail. For such a reader the book will be atrue mathematical delicatessen. I invite the reader to this delicatedinner table. Surely Zelmanov’s mathematical delicatessen will sat-isfy the requirements of all true gourmets.

D. R.

1. Zelmanov A.L. Chronometric invariants and co-moving coordinatesin the general relativity theory. Doklady Acad. Nauk USSR, 107 (6),815–818, 1956.

2. Zelmanov A.L. On the relativistic theory of anisotropic inhomoge-neous Universe. Proceedings of the 6th Soviet Conference on Cosmo-gony, Nauka, Moscow, 144–174, 1959 (in Russian).

3. Zelmanov A.L. On the statement of the problem of the infinity ofspace in the general relativity theory. Doklady Acad. Nauk USSR,124 (5), 1030–1034, 1959.

12 Foreword

4. Zelmanov A.L. On the problem of the deformation of the co-movingspace in Einstein’s theory of gravitation. Doklady Acad. Nauk USSR,135 (6), 1367–1370, 1960.

5. Zelmanov A.L. Kinemetric invariants and their relation to chrono-metric invariants in Einstein’s theory of gravitation. Doklady Acad.Nauk USSR, 209 (4), 822–825, 1973.

6. Zelmanov A.L. Orthometric form of monad formalism and its rela-tions to chronometric invariants and kinemetric invariants. DokladyAcad. Nauk USSR, 227 (1), 78–81, 1976.

7. Zelmanov A.L. and Khabibov Z.R. Chronometrically invariant var-iations in Einstein’s gravitation theory. Doklady Acad. Nauk USSR,268 (6), 1378–1381, 1982.

8. Zelmanov A.L. and Agakov V.G. Elements of the General Theoryof Relativity. Nauka, Moscow, 1989 (in Russian).

9. Cattaneo C. General Relativity: Relative standard mass, momen-tum, energy, and gravitational field in a general system of reference.Il Nuovo Cimento, 10 (2), 318–337, 1958.

10. Cattaneo C. On the energy equation for a gravitating test particle.Il Nuovo Cimento, 11 (5), 733–735, 1959.

11. Cattaneo C. Conservation laws in General Relativity. Il Nuovo Ci-mento, 13 (1), 237–240, 1959.

12. Cattaneo C. Problemes d’interpretation en Relativite Generale. Col-loques Internationaux du Centre National de la Recherche Scienti-fique, No. 170 “Fluides et champ gravitationel en Relativite Gene-rale”, Editions du Centre National de la Recherche Scientifique,Paris, 227–235, 1969.

♦

Introducing Chronometric Invariants(by the Editor)

The essence of Zelmanov’s mathematical method — the theory ofchronometric invariants — is as follows.

A regular observer perceives four-dimensional space as thethree-dimensional spatial section x0= const, pierced at each pointby time lines xi= const.∗ Therefore, physical quantities perceivedby an observer are actually projections of four-dimensional quanti-ties onto his own time line and spatial section. The spatial sectionis determined by a three-dimensional coordinate net spanning areal reference body. Time lines are determined by clocks at thosepoints where the clocks are located. If time lines are everywhereorthogonal to the spatial section, the space is known as holonomic.If not, there is a field of the space non-holonomity — the non-orthogonality of time lines to the spatial section, manifest as athree-dimensional rotation of the reference body’s space. Such aspace is said to be non-holonomic. In the general case, the space iscurved, inhomogeneous, and deforming.

By mathematical means, four-dimensional quantities can beprojected onto an observer’s time line by the projecting operator

bα =dxα

ds,

the observer’s four-dimensional velocity vector tangential to hisworld-line, while the projection onto his spatial section is made bythe operator

hαβ = −gαβ + bαbβ ,

which are satisfying to the properties bαbα=1 and hiαbα=0 requir-

ed to such projecting operators. (Other components of the tensorhαβ are: hαβ =−gαβ + bαbβ, hαβ =−g

αβ + b

αbβ.)A real observer rests with respect to his reference body (bi=0).

In other word, he accompanies to his reference body in all its

∗Greek suffixes are the space-time indices 0, 1, 2, 3, Latin ones are the spatialindices 1, 2, 3. So the space-time interval is ds2= gαβ dxαdxβ .

14 Introducing Chronometric Invariants

motions. Such person is known as accompanying observer. Project-ions of four-dimensional quantities onto the time line and spatialsection of such an accompanying observer (i. e. the observable pro-jections) are invariant in respect to transformations of time alongthe spatial section. Zelmanov therefore called such projectionschronometrically invariant quantities, i. e. “bearing the propertyof chronometric invariance”. Therefore all quantities observed bya real observer (who rests in respect to his references) are chrono-metric invariants.

So, meaning a real observer (bi=0), we have

b0 =1

√g00

, b0 = g0αbα =

√g00 , bi = giαb

α =gi0√g00

,

and also

h00 = 0 , h00 = −g00 +1

g00, h00 = 0 ,

h0i = 0 , h0i = −g0i, hi0 = δi0 = 0 ,

hi0 = 0 , hi0 = −gi0, h0i =gi0g00

,

hik = −gik +g0i g0kg00

, hik = −gik, hik = −gik = δik .

Thus, the chr.inv.-projections of a world-vector Qα are

bαQα =

Q0√g00

, hiαQα = Qi,

while chr.inv.-projections of a symmetric world-tensor of the 2ndrank, for instance Qαβ, are

bαbβQαβ =Q00g00

, hiαbβQαβ =Qi0√g00

, hiαhkβQ

αβ = Qik.

Physically observable properties of the space are derived fromthe fact that the chr.inv.-differential operators

∗∂

∂t=

1√g00

∂

∂t,

∗∂

∂xi=

∂

∂xi+1

c2vi

∗∂

∂t

are non-commutative∗∂2

∂xi ∂t−

∗∂2

∂t ∂xi=1

c2Fi

∗∂

∂t,

∗∂2

∂xi∂xk−

∗∂2

∂xk∂xi=2

c2Aik

∗∂

∂t,

and also from the fact that the chr.inv.-metric tensor hik may notbe stationary. The observable characteristics are the chr.inv.-vectorof gravitational inertial force Fi, the chr.inv.-tensor of angular ve-

Introducing Chronometric Invariants 15

locities of the space rotation Aik, and the chr.inv.-tensor of rates ofthe space deformations Dik, namely

Fi=1

√g00

(∂w

∂xi−∂vi∂t

)

,√g00=1−

w

c2

Aik=1

2

(∂vk∂xi

−∂vi∂xk

)

+1

2c2(Fivk−Fkvi) , vi=−c

g0i√g00

,

Dik=1

2

∗∂hik∂t

, Dik=−1

2

∗∂hik

∂t, Dk

k=∗∂ ln

√h

∂t,

where w is gravitational potential, vi is the linear velocity of thespace rotation,

hik = −gik +g0i g0kg00

= −gik +1

c2vivk

is the chr.inv.-metric tensor, which possesses all the properties ofthe fundamental metric tensor gαβ in the spatial section. (Hereh=det ‖hik‖, hg00=−g, while g=det ‖gαβ‖). Observable inhomo-geneity of the space is set up by the chr.inv.-Christoffel symbols

Δijk = himΔjk,m =1

2him

( ∗∂hjm∂xk

+∗∂hkm∂xj

−∗∂hjk∂xm

)

which are built just like Christoffel’s usual symbols Γαμν = gασΓμν,σ

using hik instead of gαβ. A four-dimensional generalization of themain chr.inv.-quantities Fi, Aik, and Dik (by Zelmanov) is:

Fα = −2c2bβaβα , Aαβ = chμαh

νβaμν , Dαβ = chμαh

νβdμν ,

where aαβ = 12 (∇α bβ −∇β bα), dαβ =

12 (∇α bβ +∇β bα).

In this way, for any equations obtained using general covariantmethods, we can calculate their physically observable projectionson the time line and the spatial section of any particular referencebody and formulate the projections in terms of their real physicallyobservable properties, from which we obtain equations containingonly quantities measurable in practice.

Zelmanov deduced chr.inv.-formulae for the space curvature.He followed that procedure by which the Riemann-Christoffel ten-sor was built: proceeding from the non-commutativity of the sec-ond derivatives of an arbitrary vector

∗∇i∗∇kQl−

∗∇k∗∇iQl =

2Aikc2

∗∂Ql∂t

+H...jlki∙Qj ,

16 Introducing Chronometric Invariants

he obtained the chr.inv.-tensor

H...jlki∙ =

∗∂Δjil

∂xk−

∗∂Δjkl

∂xi+ΔmilΔ

jkm−Δ

mklΔ

jim ,

which is similar to Schouten’s tensor from the theory of non-holonomic manifolds. The tensor H ...j

lki differs algebraically from theRiemann-Christoffel tensor because of the presence of the spacerotation Aik in the formula for non-commutativity. Nevertheless itsgeneralization gives the chr.inv.-tensor

Clkij =1

4(Hlkij −Hjkil +Hklji −Hiljk) ,

which possesses all the algebraic properties of the Riemann-Christoffel tensor in this three-dimensional space and, at the sametime, the property of chronometric invariance. Therefore Zelmanovcalled Ciklj the chr.inv.-curvature tensor as the tensor of the ob-servable curvature of the observer’s spatial section. Its contractionterm-by-term

Ckj = C ∙∙∙ikij∙ = himCkimj , C = C

jj = hljClj

gives the chr.inv.-scalar C, which is the observable curvature ofthis three-dimensional space. Chr.inv.-projections of the Riemann-Christoffel tensor are:

Xik=−c2R∙i∙k0∙0∙

g00, Y ijk=−c

R∙ijk0 ∙∙∙√g00

, Zijkl=c2Rijkl.

In this way, for any equations obtained using general covariantmethods, we can calculate their physically observable projectionson the time line and the spatial section of any particular referencebody and formulate the projection in terms of their real physicallyobservable properties, from which we obtain equations containingonly quantities measurable in practice.

This completes the brief introduction to Zelmanov’s mathemat-ical apparatus of chronometric invariants, that is required to betterunderstanding of the Zelmanov book.

♦

Chapter 1

PRELIMINARY NOTICES

§1.1 The initial suppositions of today’s relativistic cosmology

At the present time two main cosmological theories, referred to as“relativistic”, exist. Both are theories of a homogeneous universe.They are also known as theories of an expanding universe. One ofthe theories is derived from Einstein’s General Theory of Relativity(see, for instance, [1]), the other — from Milne’s Kinematic Theoryof Relativity [2]. Usually one calls the second of the cosmologicaltheories “kinematic”, while “relativistic” is reserved for only thefirst of them. We adhere to this terminology herein.

The General Theory of Relativity and the Kinematic Theoryof Relativity, being continuations of Einstein’s Special Theory ofRelativity, extend the Einstein theory in two different directions.From the logical viewpoint the theories exclude one another, andfrom the physical viewpoint they are absolutely inequivalent. In-equivalent also are the contemporary theories of a homogeneousuniverse — the relativistic and the kinematic ones. The first isone of possible cosmological constructions, based on a confirmedphysical theory. Future cosmological constructions, built on its base,could be theories of an inhomogeneous universe. The second is asection of the Kinematic Theory of Relativity, pretending to be aphysical theory, one of the main points of which is the cosmologicalprinciple (this principle leads to the necessity of homogeneity). Itis possible to maintain from the relativistic theory that cosmologyis deduced from physics there. On the contrary, the kinematictheory deduces physics from cosmology. Experimental disproof ofthe theory of a homogeneous universe must: (1) create the theoryof an inhomogeneous universe on the basis of the General Theoryof Relativity, in the first case; (2) overthrow the Kinematic Theoryof Relativity itself, in the second case.

We will not consider the Kinematic Theory of Relativity here.We will also have no use for the Special Theory of Relativity. So the

18 Chapter 1 Preliminary Notices

term “relativistic” will always connote a relationship to Einstein’sGeneral Theory of Relativity.

The relativistic theory of a homogeneous universe is derivedfrom the following suppositions. The first of them is:

Einstein’s equations of gravitation are applicable to the Uni-verse as a whole.

This supposition defines the cosmological theory as relativistic, be-cause the supposition considers the Einstein equations

Gμν = −κ(Tμν −

1

2gμν T

)+ Λ gμν , μ, ν = 0, 1, 2, 3 (1.1)

and their applicability at different scales. Suppositions, defining thetheory as a theory of a homogeneous universe, can be formulatedin different ways. Tolman (see [1], p. 362) merges them into theunited supposition:

If we take any point, in relation to which matter located nearthe point is at rest (on the average) at any moment of time,then observations we make at the point show that spatial di-rections are independent of one another in the “large scale” —in other words, the space is isotropic.

A mathematically more useful statement of this supposition, butless obviously physical, has been introduced by Robertson [3].

This statement of the first initial supposition contains Einstein’sequations with the cosmological constant, the numerical value ofwhich, or even its sign, is an open problem in contemporary cosmo-logy. In general, the constant can be negative, zero, or positive.The cosmological constant of a positive numerical value had beenintroduced by Einstein [4] in his theory of a static universe∗. Later,Friedmann [5] developed the theory of a non-static universe, ad-ducing arguments by which the positive sign of the cosmologicalconstant, or even its non-zero numerical value in general, could beeliminated. Therefore Einstein himself [6] subsequently excludedthe constant from the equations. However, because the relativisticgravitational equations of the second order, in their general form(for instance, see [7], p. 269), are equations (1.1), relativistic cosmo-logy regularly uses the equations with the cosmological constant.

∗Cosmology uses the term “static” to mean a metric independent of time, whilethe General Theory of Relativity usually means it to be the non-orthogonality oftime to space.

1.2 The world-metric 19

At the same time those cases where the constant is non-zero havemathematical interest rather than physical meaning.

Contemporary cosmology peculiarly identifies the Metagalaxy,supposed infinite, with the Universe as a whole. For this reasonwe use the terms “neighbourhood” and “large scale” in the secondinitial supposition∗, according to which we assume volumes elem-entary, if the volumes contain so many galaxies that matter insidethe volumes can be assumed to be continuously distributed (seealso §1.11).

§1.2 The world-metric

The second initial supposition implies the following:

Any point, in relation to which matter located in the neigh-bourhood is at rest, can be considered as the centre of a spati-ally spherical symmetry (see [1], p. 368).

This supposition, taking the relativistic equations of motion andShur’s theorem into consideration (for instance, see [8], p. 136),gives the possibility of taking a coordinate frame which, being atrest (on the average) with respect to the matter, measures cosmicuniversal time — such time satisfies the conditions

g00 = 1 , g0i = 0 , i = 1, 2, 3 . (2.1)

The space of this coordinate frame is a constant curvature spaceof curvature 1

3C. This space, undergoing homologous expansionsand contractions with

C = 3k

R2, k = 0, ±1 , R = R (t) , (2.2)

in conformally Euclidean spatial coordinates has the metric

ds2 = c2dt2 −R2dx2 + dy2 + dz2

[1 + k

4(x2 + y2 + z2)

]2 . (2.3)

The case of k=+1 had first been considered by Einstein [4] forhis static model, and by Friedman [5] for the non-static models.The case of k=0 had first been considered by de Sitter [10] for the

∗We use the terms “neighbourhood” and “large scale” in this sense throughout,unless otherwise stated.


empty static model, by Lemaıtre [11] for the non-empty non-staticmodel, and by Robertson [3] for the non-empty models. The caseof k=−1 had first been considered by Friedmann [12].

It is known from geometry that a space of k=+1 is locallyspherical, a space of k=0 is locally Euclidean, and a space of k=−1is locally hyperbolic. Taking spherical symmetry with respect toany point, and applying to it the properties of coherence, we conclu-de that the space of k=+1 is elliptic (actually, doubly-connected) orspherical (actually, simply connected), and the spaces of k=0 andof k=−1 are Euclidean and hyperbolic respectively (both spacesare infinite and simply connected).

§1.3 Properties of matter

Taking gαβ from (2.3) and substituting it into the Einstein equationsof gravitation

Gμν −1

2gμνG = −κTμν− Λ gμν , (3.1)

we deduce that only two functions of time, namely — the functions

ρ = ρ (t) , p = p (t) , (3.2)

exist, where Tμν can be expressed by the formulae

T 00 =1

g00

(ρ+

p

c2

)−p

c2g00, (3.3)

T 0j = −p

c2g0j , (3.4)

T ik = −p

c2gik, (3.5)

or, in other words,

Tμν =(ρ+

p

c2

) dxμ

ds

dxν

ds−p

c2gμν , (3.6)

wheredxσ

ds=

1√g00

, 0, 0, 0 (3.7)

is a four-dimensional velocity, which characterizes the mean mo-tion of matter in the neighbourhood of every point (irregular de-viations from the mean motion are taken into account with the

1.4 The law of energy 21

scale parameter p). Formula (3.6) coincides with the formula for theenergy-momentum tensor of an ideal fluid∗, the density of whichis ρ and the pressure is p. Hence matter, in the idealized universewe are considering, can be considered as an ideal fluid, which,having a homogeneous density and pressure in accordance with(3.2), is at rest with respect to the non-static space. In other words,the ideal fluid undergoes expansions/contractions, accompanied bythe non-static space. The pioneering work in relativistic cosmologyundertaken by Einstein and Friedmann†, did not take the pressureinto account. Friedmann was the first to consider the static models,so the case of ρ> 0, p=0 is known as Friedmann’s case of aninhomogeneous universe. The first to introduce p> 0, was Lemaıt-re [15, 16].

§1.4 The law of energy

One of the consequences, which could be deduced from the Ein-stein equations of gravitation, is the relativistic law of energy

∂T νμ∂xν

− Γσμν Tνσ +

∂ ln√−g

∂xσT σμ = 0 . (4.1)

As a result of (2.3) and (3.3–3.5), the equations (4.1) with μ=0take the form

ρ+ 3R

R

(ρ+

p

c2

)= 0 , (4.2)

where the dot denotes differentiation with respect to time. Theequations (4.1) with μ=1, 2, 3 become the identities

0 = 0 . (4.3)

Equation (4.2) is one of the main equations of the relativistictheory of a homogeneous universe (the cosmological equations). Wewill refer to this equation as the cosmological equation of energy.

Let us take any fixed volume of space, i. e. a space volumelimited by surfaces, equations of which are independent of time.We can write the value V of such a volume as follows

V = R3Π0 , Π0 6 ‖ t , (4.4)

∗Ideal in the sense of the absence of viscosity, not in the sense of incompress-ibility.

†We do not take the works of de Sitter [9, 10], Lanczos [13], Weyl [14], andLemaıtre [11] into account, because they considered the empty models.


then the energy E inside this volume is

E = ρR3Π0c2. (4.5)

As a result of (4.4) and (4.5), we can transform the cosmologicalequation of energy (4.2) into the form

dE + pdV = 0 , (4.6)

which is the condition for adiabatic expansion and contraction ofthe space.

In the Friedmann case, where

p = 0 , (4.7)

it is evident thatdE

dt= 0 , (4.8)

and becauseE =Mc2, (4.9)

where M is the mass of a matter inside the volume V , we have

dM

dt= 0 . (4.10)

So in the absence of pressure the mass and the energy of anyfixed volume remain unchanged.

§1.5 The law of gravitation

As a result of (2.3) and (3.3–3.5), the equations of the relativisticlaw of gravitation with μ, ν =0, with μ=0, ν=1, 2, 3, and with μ, ν ==1, 2, 3 give the following formulae, respectively: the equation

3R

c2R= −

κ

2

(ρ+ 3

p

c2

)+ Λ , (5.1)

the identity0 = 0 , (5.2)

and the equation

R

c2R+ 2

R2

c2R2+ 2

k

R2=κ

2

(ρ−

p

c2

)+ Λ . (5.3)

Eliminating R from (5.1) and (5.3), we obtain

3R2

c2R2+ 3

k

R2= κρ+ Λ . (5.4)

1.6 The system of the cosmological equations 23

Equations (5.1) and (5.3), as well as (5.4) and (4.2), are the mainequations of the relativistic theory of a homogeneous universe (thecosmological equations). So, we will refer to the equations (5.1)and (5.3), and also to their consequence (5.4), as the cosmologicalequations of gravitation.

It is easy to see that the cosmological equation of energy is aconsequence of the cosmological equations of gravitation.

§1.6 The system of the cosmological equations

There are only two independent equations of the cosmological equa-tions, namely (4.2) and (5.4)

ρ+ 3R

R

(ρ+

p

c2

)= 0

3R2

c2R2+ 3

k

R2= κρ+ Λ

. (6.1)

The remaining equations can be deduced as their consequences.So we have a system of two equations, where three functions areindependent, namely, the functions

ρ = ρ (t) , p = p (t) , R = R (t) . (6.2)

To find the functions we need to add a third independent equa-tion to the system. It could be, for instance, the equation of state,which links density and pressure. A form of the curves themselves(6.2) could be found without the third equation, if we were to makevalid limitations on density and pressure. Usually one studies thecurve of the third function of (6.2) under the supposition that

ρ > 0 , p > 0 ,dp

dR6 0 . (6.3)

On the one hand, for any substance, we have

T > 0 , (6.4)

and also for radiations (including electromagnetic fields) we have

T = 0 . (6.5)

On the other hand, for ideal fluids (3.6) we have

T = ρ− 3p

c2. (6.6)


So, in general, we have

ρ > 3p

c2. (6.7)

Therefore, instead of (6.3), we can write

ρ > 3p

c2, p > 0 ,

dp

dR6 0 . (6.8)

We will limit our tasks here, because we will consider only non-empty models — those models, where

ρ > 0 . (6.9)

§1.7 Its solutions. The main peculiarities.

The main properties of solutions like R=R (t) can be routinelyfound for different Λ and k, by studying nulls of the functions

f (R,Λ) = R2 =κ

2ρc2R2 +

Λ

3R2 − k , (7.1)

see [5, 12, 17, 18] and [1], p. 359–405, for instance.In this case one makes a tacit supposition (which is physically

reasonable) that the essential positive function R (t) is continuouseverywhere, where the function exists. Moreover, one supposes itsderivative continuous under R 6=0 (besides, see §1.13). The mainresults the case ρ> 0 (6.9) give will be discussed in the next section.We will show here only several of the main properties of the func-tion R (t), obtained in another way, similar to the way we willconsider in §4.19–§4.23. First of all, it is evident that numericalvalues of R, between which the function R (t) (under the afore-mentioned supposition) is monotone, are:

(1) zero;

(2) finite minimal values;

(3) finite values, which are asymptotic approximations of R fromabove or from below (under the conditions t→+∞ or t→−∞);

(4) finite maximal values;

(5) infinity.

For all the finite values of R we write respectively, instead of(3.1) and (3.4),

3R = −κ

2

(ρc2 + 3p

)R+ Λc2R , (7.2)

3R2 + 3kc2 = κρR2 + Λc2R2. (7.3)

1.7 Its solutions. The main peculiarities. 25

It is seen from (4.4–4.8) that the density changes as fast as R−3

under zero pressure. It changes faster than R−3 when the pressurebecomes positive. Note also, because of the first and the secondconditions (6.8), we have

ρR 6(ρ+ 3

p

c2

)R 6 2ρR , (7.4)

so, if ρR approaches zero or infinity, then(ρ+3

pc2

)R also ap-

proaches zero or infinity, respectively.Similarly, in the case of the density changes (7.3), it follows

that when R approaches zero, ρ and R2 approach infinity — themodel evolves into the special state of infinite density (R2=∞). Asseen from (7.2) and (7.3), this case is possible under any numericalvalues of Λ and k.

From the aforementioned density changes we can also conclude,when R approaches infinity, then the density and the pressurerespectively approach zero — the model evolves into the ultimatestate of the infinite rarefaction. Looking at (7.2) and (7.3) we notethat the abovementioned evolution scenario is possible with Λ> 0under any numerical values of k, with Λ=0 under only k=0 ork=−1, and the scenario is impossible with Λ< 0.

When R transits its minimum value, i. e. when the model tran-sits the state of minimum volume, then R becomes zero and R isnonnegative. When R approaches a non-minimum numerical value,i. e. when the model evolves asymptotically to a static state, then Rand R approach zero. It is evident that if R is static, the model willalso be static. The formulae (7.2) and (7.3) for all the three casesgive

3k > ΛR2, (7.5)

0 < ΛR . (7.6)

From this we can conclude that the transit of the model throughthe state of minimum volume, the asymptotic approach of themodel to a static state, and the static models in general, can occuronly when Λ> 0, k=+1.

When R transits its maximum numerical value, i. e. when themodel evolves through the state of its maximum volume, then Rbecomes zero and R is nonpositive. Then, by formulae (7.2) and(7.3), we obtain

3k > ΛR2, (7.7)

0 TΛR . (7.8)


So the transit of the model through the state of its maximumvolume is possible with Λ> 0 only for k=+1, and it is possible forΛ< 0 under any numerical values of k.

Let us suppose that the model undergoes monotone transforma-tions of its volume. The transformations are limited by the stateof the minimum volume or a static state from below, and they arelimited by the state of the maximal volume or another static statefrom above. We will mark the lower and the upper ultimate statesby indices 1 and 2, respectively. Then

R1 > 0 > R2 , R1 < R2 , ρ1 > ρ2 , (7.9)

and formula (5.1) givesp1 < p2 , (7.10)

which contradicts the third of the conditions (6.8). So the supposedkinds of the evolution of the model are impossible (see also §1.13).

§1.8 Types of non-empty universes

Let us make a list of the kinds of non-empty universes accordingto the contemporary classification of cosmological models∗ (for in-stance, see [18]).

Static models or, in other words, Einstein’s models (Type E)

In this case R remains unchanged. Equations (5.1) and (5.4) give,respectively

κ

2

(ρ+ 3

p

c2

)= Λ , (8.1)

3k

R2= κρ+ Λ , (8.2)

consequences of which are that ρ and p remain unchanged as well,and that Λ> 0, k=+1.

The Einstein models are unstable. Consequently, their fluctua-tions, their homogeneity unchanged, alter one or two of the quan-tities R, ρ, and p, so the models compress themselves up to thespecial state of infinite density, or, alternatively, up to the ultimatestate of infinite rarefaction. Eddington [19] was the first to discoverthe instability of the Einstein models (as a matter of fact, theinstability is evident if you consider Friedmann’s equations [5]).

∗This classification, in its main properties, is in accordance with Friedmann [5].

1.8 Types of non-empty universes 27

Models of the first kind — asymptotic, monotone, and oscillatingmodels

All the models evolve through the special state of infinite density.Asymptotic models of the first kind (type A1) change their vol-

umes monotonically during their expansions or contractions be-tween the special state of infinite density, when time has a finitevalue t= t0, and a static state, corresponding to the Einstein modelwhen t=+∞ or t=−∞. The aforementioned models are possibleonly when Λ> 0, k=+1.

Monotone models of the first kind (type M1) change their vol-umes monotonically during their expansions or contractions be-tween the special state of infinite density, when time has a finitevalue t= t0, and the ultimate state of infinite rarefaction, whent=+∞ or t=−∞. The models are possible with Λ> 0 for any num-erical value of k, and with Λ=0 for k=0, k=−1.

Oscillating models of the first kind (type O1) expand their vol-umes from the special state of infinite density, when t= t1, and theycontinue expanding up to their maximum volumes, when t= t0.Then they contract themselves into the special state of infinitedensity, reaching that state when t= t2 (t1, t0, t2 have finite values).The models are possible with Λ> 0 for k=+1, or with Λ< 0 underany numerical value of k.

When a model of any of the aforementioned kinds evolvesthrough the special state, three cases are possible:

(a) the type of model remains unchanged (it is always known forp=0);

(b) the type of the model changes from one to another; at thesame time the new kind will also be of the first kind;

(c) the numerical value of R, which is usually real, becomes im-aginary (R2 changes its sign).

Models of the second kind — asymptotic and monotone ones∗

The models do not transit through the special state of infinitedensity.

Asymptotic models of the second kind (type A2) change theirvolumes during their monotone expansions or contractions betweena static state, corresponding to the Einstein model, when t=∓∞,

∗Read about oscillating models of the second kind in §1.13.


and the ultimate state of infinite rarefaction, when t=±∞. Themodels are possible only for Λ> 0 and k=+1.

Monotone models of the second kind (type M2) contract theirvolumes from the ultimate state of pure vacuum, when t=−∞,up to the state of their maximum volumes, when time has a finitevalue t= t0. Then the models go into expansion again up to theultimate state of infinite rarefaction, when t=+∞. The models arepossible only for Λ> 0 and k=+1.

Table 1.1 shows which kinds of non-static non-empty universesare possible under different numerical values of Λ and k.

k = +1 k = 0 k = −1

Λ > 0 A1 M1 O1 M1 M1

A2 M2

Λ = 0 O1 M1 M1

Λ < 0 O1 O1 O1

Table 1.1 Types of the models of non-staticnon-empty universes.

In the cases of p> 0 or p=0, the possible kinds of the modelsand their locations in the table’s cells are the same.

§1.9 Sections of relativistic cosmology

Considering contemporary cosmology, we select five main prob-lems, which define its main sections. Three of them define sectionsof the relativistic theory itself — the dynamics, the three-dimen-sional geometry, and the thermodynamics of models of the Uni-verse. The fourth links the relativistic theory to Classic Mechanics.The fifth compares the theory with observational data.

The main part of relativistic cosmology is the dynamics of theUniverse. It consists of: (1) studies of the evolution of the models,the main results of which are given in §1.7–§1.8; (2) studies of thestability of the models (this problem is not actually in the literatureyet, besides the aforementioned stability of the Einstein models, see§1.8). The dynamics borders on the space geometry from one side,and on the thermodynamics from the other side.

Related to the space geometry are the questions: which are thespace curvature and the topological structure of the space? Themain results of the studies are given in that part of §1.2, which isrelated to the properties of three-dimensional space.

1.9 Sections of relativistic cosmology 29

The thermodynamics of the relativistic homogeneous universeis historically associated with Tolman, who introduced the relativ-istic generalization of Classical Thermodynamics [1]. The first lawof Tolman’s relativistic thermodynamics is, naturally, the law ofthe conservation of energy (see [20] or [1], p. 292), one of the formsof which are equations (4.1). The second law of thermodynamics(see [21] or [1], p. 293) can be represented in the form

1√−g

∂

∂xν

(

ϕdxν

ds

√−g

)

dΣ >dQ

Θ, (9.1)

where ϕ and dxν

dsare the entropy density and the for-dimensional

velocity of an elementary four-dimensional volume dΣ, respect-ively. The quantity dQ here is the gain of heat between the timeborders of the volume through its spatial borders (Θ is their ownabsolute temperature). The inequality sign here is related, as itis evident, to reversible processes. The equality sign is related toreversible processes. In the case of the cosmological models we areconsidering, the first law leads to the condition of an adiabatic state(4.6). Under this condition, we can clearly see that there is no gainof heat in the model (it is also a result of the initial supposition thatthe model is isotropic, see §1.1). Taking it into consideration in aspace of the metric (2.3), the second law of thermodynamics leadsto the condition that the entropy of any fixed (in the sense of §1.4)volume does not decrease (compare with [1], p. 424)

∂

∂t(ϕV ) > 0 . (9.2)

Matter in the model is being considered in general as a mixtureof two interacting components — a pressured substance and iso-tropic radiations. In particular cases, interactions between the twocomponents or the pressure of the substance can be absent.

Let us mention another of Tolman’s main results related tothe cosmological models we are considering. Arising from the factthat the models do not contain heat fluxes, friction, or pressuregradients, Tolman discovered that expansions and contractions ofthe models can be reversible (see [22] and [23]; and also [1], p. 322,p. 426 etc.). This does not contradict models containing radiation,since every observer who is at rest can detect a flux of the radiationthrough his environment’s surface, of an arbitrary constant radius— from within under the expansions, and from outside under thecontractions of the model (see [1], p. 432 etc.). Sources of irreversi-


bility can be contained in only physical chemical processes withinevery element of the matter. The reversible expansions must ac-company transformations of a portion of the substance into ra-diation (a portion of the substance is annihilated), the reversiblecontractions must accompany the inverse process — the reconsti-tution of the substance (see [23] or [1], p. 434). Finally, becauseevery fixed (in the sense of §1.4) volume has no constant energy,the cosmological model has no fixed maximum entropy, so theirreversible processes can be infinite — the model does not evolveinto a state of the maximum entropy (see [24] or [1], p. 326 etc.,p. 439 etc.).

Relativistic cosmology is linked to Classical Mechanics by theanalogy between the relativistic cosmological equations of a homo-geneous universe and the classical equations of a homologous ex-panding/contracting gravitating homogeneous sphere of an arbitr-ary radius. This analogy can apply only for the case of an idealfluid — it has no tensions or pressure (p=0). The analogy was firstcreated by Milne in the case of Λ=0, k=0 (see [25] or [2], p. 304).Then McCrea and Milne studied the analogy in the cases of Λ=0,k 6=0 (see [26] or [2], p. 311). Finally, it was generalized for the caseof Λ 6=0 (see [2], p. 319). In the last case, Newton’s law of gravitationmust be generalized by introducing an additional force, producingan additional relative acceleration of two interacting particles (theacceleration equals the product of 1

3Λc2 and the distance between

the particles). If r, θ, and ϕ are the polar coordinates of an arbitrarypoint on the spherical mass we are considering, where the origin ofthe coordinates can be fixed at any other point of the sphere (ρ isits density, γ is Gauss’ constant, and ε is the integrating constant),then in that case we obtain the condition of homology

r

r= f (t) 6 ‖ r, θ, ϕ , (9.3)

the condition of continuity

ρ+ 3r

rρ = 0 , (9.4)

the equation of motion (in the form, transformed for the small)

3r

r= −4πγρ+ Λc2, (9.5)

and the integral of energy

3r2

r2− 6

ε

r2= 8πγρ+ Λc2. (9.6)

1.10 Accompanying space 31

Introducing the real function of time R (t), which satisfies theconditions

R

R=r

r,

k

R2= −2

ε

c2r2, k = 0,±1 , (9.7)

and taking into account

κ =8πγ

c2, (9.8)

we transform (9.4), (9.5), and (9.6) to the form, which is identicalto equations (4.2), (5.1), and (5.4) in the case of p=0. It is evidentthat the analogy we have considered facilitates determination ofa link between the dynamics and the geometry of the relativistichomogeneous universe — the link between the evolution scenarioof the models and the space curvature.

The question “Does the cosmological theory correlate to obser-vational data?” remains. We reserve this question for §1.14.

§1.10 Accompanying space

As we have seen, the three-dimensional space in which the relativ-istic theory of a homogeneous universe operates is an accompanyingspace — a space which moves in company with matter at any of itspoints. Therefore the matter is at rest (on the average) with respectof the space. So expansions or contractions of the space imply theanalogous expansions or contractions of the matter itself.

It is evident that the accompanying space is primary from thephysical viewpoint. For this reason determining the geometricproperties of such a space is of the greatest importance. In par-ticular, the question “Is the space infinite, or not?” is the same asthe question “Is the real Universe spatially infinite, or not?”, in areasonable physical sense of the words. Geometric characteristicsof the accompanying space are directly linked to observational data,because the data characterize cosmic objects, which are at rest (onthe average) with respect of the space.

We can say that relativistic cosmology studies relative motionsof the elements of matter as the analogous motions of the elementsof the accompanying space, and deformations of matter are con-sidered as the analogous deformations of the accompanying space.

The idea of studying motions in this way is very simple. In fact,let us consider, for instance, two points

x = α , x = β , (10.1)


located on the x axis. The distance between the points along theaxis can be expressed by the integral

J =

∫ β

α

√a dx , (10.2)

where a is the necessary coefficient of the spatial quadratic form.In the coordinate frame, which accompanies the points (10.1), αand β are constant, so the motion of the points with respect to oneanother along the x-axis will be described by the function

a = a (t) . (10.3)

This method of studying motions is conceivable, of course, inClassical Mechanics too. At the same time in Classical Mechanicswe have: (1) this method studying motions, as well as the standardstudies of motions with respect to the static space of ClassicalMechanics, as the final result must be based on equations of motion;(2) the geometrical properties of the accompanying space are thesame as the properties of the static space. On the contrary, in therelativistic theory we have: (1) this method of studying deforma-tions of the accompanying space realizes itself by the equations ofgravitation, the equations of motion remain unused; (2) in general,the geometric properties of the accompanying space are differentfrom the properties of any other space, moreover the static spacecan be introduced in only an infinitesimal location.

§1.11 The cosmological equations: the regular interpretation

From the usual viewpoint the cosmological equations are consider-ed as the equations of the whole Universe, defining its “radius” R.From this viewpoint∗ the quantities

C = 3k

R2, (11.1)

D = 3R

R(11.2)

are the tripled curvature and the relative rate of the volume expan-sion of the accompanying space as a whole. The quantities ρ and p

∗Note, one sometimes means by “the radius of the Universe”, the radius of the

space curvature which, as is evident, equals R√k

.

1.11 The cosmological equations: the regular interpretation 33

are the average density and the average pressure of matter in theUniverse. From such a viewpoint, of course, the next suppositionis essential:

Supposition A The Einstein equations are applicable to the wholeUniverse.

This supposition, of course, is a far-reaching extrapolation (forinstance, see [1], p. 331). At the same time, building the theoryof the world as a whole, the supposition cannot be replaced byanother now, because the others have proved less fruitful than it atthis time.

When we, from this standpoint, apply the homogeneous cosmo-logical equations to the real Universe, we suppose, first, that:

Supposition B If we consider the Universe in the frames of thelarge scale, we can assume the Universe homogeneous.

In this case, of course, in general, it is not supposed that this scalecoincides with that scale we have mentioned in §1.1. This suppo-sition cannot be justified by any theoretical suppositions (physicalor astronomical, or otherwise), at least at this time. Moreover, itis unknown as to whether the homogeneous state of the wholeUniverse is stable or not (for instance, see [1], p. 482). Applyingthe homogeneous models to the real Universe in any special case,we need to set forth a scale, starting from which the Universewill be supposed homogeneous. Contemporary applications of theaforementioned models to the real Universe identify the Universewith the Metagalaxy. In this case it is supposed:

Supposition C The Metagalaxy contains the same quantity of mat-ter inside volumes less than the sphere of radius 108 parsecsand more than the sphere of radius 105 parsecs∗.

Next, the contemporary applications of the relativistic models tothe real Universe suppose that:

Supposition D The red shift in the spectra of extragalactic nebulae,proportional (at least, in the first approximation) to the dis-tances between them and our Galaxy, is a result of mutual ga-lactic recession (of the expansion of the accompanying space).

∗The radius 108 parsecs is in order of the distances up to the weakest in theirlight intensity of the extragalactic nebulae, observed with the 100in reflector. Theradius 105 parsecs is in order of distances between the galaxies nearest to oneanother. It is evident that “elementary” must be supposed volumes of the order of1020 cubic parsecs.


Finally, the contemporary applications of the relativistic models tothe real Universe suppose:

Supposition E Almost all the mass of the Universe is concentratedin galaxies. Alternatively, at least, to find the average densityof matter in the Universe it is enough to take into accountonly the masses of the galaxies.

Note that suppositions A and B are sufficient for deducing thecosmological equations (compare with §1.1). Suppositions C, D, andE play a part in comparing (in the qualitative comparison especially)the theory and observations. Supposition B is covered by Supposi-tion C. The last supposition is a powered expression of the sampleprinciple (for instance, see [1], p. 363 and [2], p. 123), which is veryfuzzy. This principle, specific for contemporary cosmology, sets upthe known part of the Universe as a sufficient sample for studyingthe main properties of the whole Universe.

§1.12 Numerous drawbacks of homogeneous models

The contemporary relativistic cosmology has numerous advantagesin relation to pre-relativistic cosmological theories. In particular,the contemporary theory is free of both of the classical paradoxes,namely — the photometric∗ and the gravitational), which are ir-removable in pre-relativistic cosmology if the average density ofmatter in the Universe is non-zero†. So the contemporary theoryeliminates the heat death of the Universe, because it does notconsider the contemporary state of the known fragment of theUniverse as a fluctuation. Finally, the theory naturally explains thered shift, because expansions or contractsions of the accompanyingspace are inevitable (static states of the Universe are unstable). Inparticular, it provides a possibility of linking the expansions of thespace to the fact that the substance we observe transforms itselfpartially into radiations‡.

Besides its advantages, the contemporary relativistic cosmology

∗The contemporary theory is free of this paradox, because the finite pressure ofradiations in the Universe results in the finite brightness of the sky we observe.

†Extrapolating the red shift up to infinity, we can also remove the photometricparadox under finite densities in non-relativistic cosmology. At the same time itcreates the problem of the nature of the red shift itself.

‡Eigenson’s non-relativistic theory [27] sets forth the same link, howeverhis theory requires an excessive loss of stellar masses; the stars transform intoradiations.

1.12 Numerous drawbacks of homogeneous models 35

has numerous drawbacks, which indicate that the theory is inappli-cable to the real Universe. Some of the drawbacks were mentionedin the previous §1.11, where we discussed Suppositions A and B.Other the drawbacks (they will be discussed in §1.14 and §1.15)are derived from comparison between the theory and observationaldata. Finally, the remaining drawbacks can be formulated as nota-tions on those properties of the models which are unreasonablefrom their physical sense or from more general considerations.As examples of such properties of the cosmological models, wementioning the follows:

1. The special state of infinite density. From the physical view-point, it is senseless to consider the real Universe under tran-sit through this state. So it is senseless to apply the cosmo-logical models to the real Universe under those conditions;

2. To consider the contemporary state of the Universe we ob-serve as exceptional. This point, as we will see in the nextsection, is a peculiarity of all the asymptotic and monotonemodels (see [1], p. 399 etc.);

3. To suppose the cosmological constant non-zero, having physi-cally unproved suppositions as a basis. Eddington (see [28],Chapter XIV) attempted to join relativistic cosmology andQuantum Mechanics. He attempted to show that the valueof the cosmological constant is positive Λ> 0, k=+1 in thiscase, so the real Universe must be of the type A2. Howeverhis speculations are very artificial and questionable;

4. The finiteness (the closure) of the space is a drawback fromthe general viewpoint. The supposition that space is closed,as Einstein and Friedmann made in their first models, hadbeen recognized by them as artificial after the theory of anon-static infinite universe had been created [12, 29].

As we can see from §1.7 and §1.8 (for instance, see Table 1.1 inp. 28), there are no non-static models∗ which would be free of allthe aforementioned properties, or even of three of them. Naturally:

• All the models, which are free of the 1st property (the modelsof the second kind), possess all the remaining three properties

A2M2

}

Λ > 0 , k = +1 ; (12.1)

∗We exclude the Einstein models from consideration, because of their instability.


• All the models, which are free of the 2nd property (the os-cillating models), have the 1st property and also one of theothers — the 3rd or the 4th property

O1

{Λ > 0 , k = +1

Λ < 0 , k = 0,±1

}

; (12.2)

• All the models, which are free of the 3rd property (the modelswith Λ=0), have the 1st property and also one of the others— the 2nd or the 4th property

Λ = 0

{k = +1 , O1k = 0,−1 , M1

}

; (12.3)

• All the models, which are free of the 4th property (the spa-tially infinite models), have the 1st property and also one ofthe others — the 2nd or the 3rd property

k = 0,−1

{Λ > 0 , M1

Λ < 0 , O1

}

. (12.4)

§1.13 Some peculiarities of the evolution of the models

Let us consider some peculiarities of the evolution of the homogene-ous models we have enunciated in §1.12.

Essential is that R, and consequently any fixed volume, becomeszero under the special state of infinite density. Let us consider thelower border of the changes of R, where R has a break, however Ritself is not zero (see §1.7), reaching a numerical value Rs> 0. Thenwe can assume that ρ has a singularity, namely, it becomes infinitewhen R=Rs> 0 (a positive pressure is necessary, see [3] p. 72 andp. 82, or [1] p. 399 etc.).

Examining (7.3) we can see that infinite density under finite R

requires infinite R2 and, hence, infinite RR

. The latter implies that

two infinitesimally close points of the accompanying space moveone with respect to one another with the velocity of light. At thesame time, we know that the masses of all particles become infiniteat the velocity of light. So any finite volume acquires infinite den-sity at the lower border of its changes (Rs> 0).

Considering the lower condition Rs> 0, we obtain the special

1.13 Some peculiarities of the evolution of the models 37

state of infinite super-density

ρ =∞ ,

∣∣∣∣R

R

∣∣∣∣ =∞ , (13.1)

where, to remove R=0 with Rs> 0 in the lower limit of the changesof R, does not give any advantage.

According to Einstein’s opinion [6], which has been acceptedby other scientists, the special states of infinite density can be acriterion for which some idealizations, such as the homogeneity ofthe Universe under the conditions of its maximum contraction, areabsolutely inapplicable.

Looking at the observable state of the Universe from the view-point of the asymptotic and monotone models, we can say that the

exclusivity of the state, where ρ and RR

have finite numerical values

(see §1.14), consists of the following. We assume ε1 and ε2 any infi-

nitesimal values of the quantities ρ and∣∣∣ RR

∣∣∣, respectively. Then that

time interval, during which the conditions

ρ > ε1 ,

∣∣∣∣R

R

∣∣∣∣ > ε2 (13.2)

are true, is infinite in any of the aforementioned models. Thereforewe can say of the asymptotic and homogeneous models:

Supposing that observable properties of the Universe we knowfrom its known fragment are law for all space, we conclude thatthe asymptotic and homogeneous models are rare and uniquesituations in time.

As we saw it in §1.12, no homogeneous model, which would be freeof the 1st and the 2nd properties we have mentioned above, exist.The models with the properties could be the oscillating modelsof the 2nd kind ( type O2), which change R between the finiteminimum and the finite maximum. Such evolution of R, accordingto Tolman (see [30] or [1], p. 401 etc.), is possible for Λ> 0, k=+1 ifthe third of the conditions (4.8) is violated in the contractions andthe expansions. However, he found (see [1], p. 402, 430 etc.) that thesaid violations of the condition (6.8), and hence also the possibilitythat the homogeneous models of kind O2 can exist, cannot be justi-fied from the physical viewpoint. Moreover, the models have alsothe 3rd and the 4th properties anyhow.

Assuming Λ 6=0, we obtain more possible homogeneous modelsthan it would possible for Λ=0. For instance, we can obtain the


models which, being free of the 1st property or of the 2nd and 4thproperties in their sum, are impossible under Λ=0.

Similar considerations stimulate one to consider the cases ofΛ 6=0 (for instance, see [16]).

If we wish to obtain models which are free of the 1st propertyor of the 2nd and the 3th properties, then we arrive at the modelswith k=+1. Let us recall the final remarks in §1.2 in relation toit. In accordance with §1.2, the positive space curvature leads tothe closure of space, necessary in only those models which aresymmetric with respect of any of their points (the homogeneousmodels). In general, the relation between the curvature and thecoherence properties, consisting of many more factors, has muchgreater possibilities.

§1.14 The homogeneous models and observational data

The red shift discovered by Slipher (see [10] and [7], p. 301 etc.)increases with distance and was initially considered in relation tothe empty models [10, 31, 14, 11]. Lemaıtre [15] began to considerthe non-empty non-static models in the time between two events,which suggested qualitative grounds for them, namely, Hubble’sdiscovery that galaxies are approximate distributed homogeneousin space [32], and Hubble’s discovery that red shift is approximatelyproportional to distance [33, 34].

The observational data he used gave numerical bounds of thefollowing quantities:

• Using Supposition E (see §1.11), the average density of matterin the Universe, ρ∼ 10−31 gram×cm−3. In accordance withthe late bounding, the average density is ∼ 10−30 gram×cm−3

[35, 36];

• Using Supposition D, the data gave the relative speed of linear

expansions of space, RR= 1.8×10−17 sec−1, as the factor of pro-

portionality between the expansion speed and the distance (inthis bounding, he used measurements of the absolute bright-ness of all galaxies and their number in a unit of volume).

Those bounds, in connection with some of other bounds, lead tothe conclusion that in the contemporary epoch (even if we rejectSupposition E),

p� ρc2 , κ p�R2

R2, (14.1)

1.14 The homogeneous models and observational data 39

hence, using the cosmological equations for numerical calculations,we can set p=0.

Moreover, as it is easy to see,

3R2

c2R2> κρ . (14.2)

Therefore, the following cases are possible in a homogeneousuniverse∗

Λ > 0

{k = +1

k = 0,−1 ,

A1 , M1 , M2

M1

Λ = 0 , k = −1 , M1

Λ < 0 , k = −1 , O1

. (14.3)

To draw conclusions which would be more likely than the above,and to make quantitative tests of the relativistic cosmological equa-tions, more detailed observational data are needed on one hand and,on the other hand, some additional theoretical correlations betweenthe observable values we use in the cosmological models.

More detailed statistical data had been collected by Hubble [37].Those data concern:

(a) the average spectral type of galaxies;

(b) the numerical values of the red shift δ= Mλλ

in the spectra of

the galaxies; their photographic stellar magnitudes go up tom=17;

(c) the number of the galaxies N(m), absolute stellar magnitudesof which go up to m= 21.

From the necessary theoretical correlations, we select the fol-lowing

Φ1

{lm , m , δ ; T

}= 0 , (14.4)

Φ2

{

δ , lm ;R

R,R

R, . . .

}

= 0 , (14.5)

Φ3

{

N(lm) , lm ; n ,k

R2

}

= 0 , (14.6)

∗These are true, because the cosmological equations under conditions (14.1),(14.2) with Λ> 0, k=+1 become R> 0 (which is impossible in the models A1and O1).


and also

Φ4

{

N(δ) , δ ; n ,R

R,R

R, . . .

}

= 0 , (14.7)

using which, we obtain

Φ5

{dN

dδ, δ ; n ,

R

R,R

R, . . .

}

= 0 . (14.8)

Photometric distances lm here are the distances we calculate,using the inverse square law in photometry, from the stellar magni-tudes m corrected with the red shift in accordance with (14.4)∗. Theexplicit temperature T can characterize a galaxy, if its spectrum isapproximated to Planck’s spectrum. The number of galaxies in anunit of volume is n. The number of the galaxies, which range up tostellar magnitude m, is N(lm). The number of galaxies which rangeup to the numerical value δ of their red shift, is N(δ). Numerical va-lues of all the quantities are taken in the epoch of the observations.

It is possible to compare the theory with the observations indifferent ways. Let us consider two of them:

1. Assuming the numerical value T ≈ 6000◦ the data (a) gave,employing (14.4), we can compare the data (b) with formula(14.5). As a result we can obtain, besides the known numerical

value of RR

, the numerical value of RR

. Next, using (14.4) againand extrapolating the data (b) up to m= 21, i. e. for all thearea of the data (c), we can compare the data (c) with (14.6).

A result will be the numerical value of kR2

. Substituting theobtained values into the cosmological equations (5.1) and (5.4),we can obtain the numerical values of Λ and ρ. The value of ρ,as it is easy to see, will be obtained without Supposition E. Sowe can compare ρ with its known numerical value, obtainedin the framework of Supposition E;

2. Extrapolating the data (b) for all data (c) and eliminating m,we can compare the results with (14.7) and (14.8). A result

will be, besides the numerical values of RR

and ρ we knew be-

fore, the numerical value of RR

. Substituting the values into

the cosmological equations (5.1) and (5.4), we can obtain the

numerical values of Λ and kR2

. Using the last value, we can

∗In non-Euclidean spaces and non-static spaces, the photometric distances arethe same as regular distances in only the infinitesimal scale.

1.14 The homogeneous models and observational data 41

compare (14.4) and (14.6) with the data (b). We can also obtainthe numerical value of T , which is in (14.4). This numericalvalue can be compared with the data (a).

Hubble [37] followed the first approach. He used a method de-veloped by himself and Tolman [38]. Using the assumed numerical

value T ≈ 6000◦, he found the values of Λ and kR2

to be positive for

R≈ 1.45×108 parsecs (this value of R is in the order of the radius ofthe contemporary volume of the space). He had also obtained thelarge value ρ≈ 6×10−27 gram×cm−3 and the type M1.

The second approach had been realised in McVittie’s works∗,who used McCrea’s formulae [43]. Assuming ρ∼ 10−30 gram×cm−3,

he obtained negative values of Λ and kR2

negative (he assumed

R∼ 108–109 parsecs), the Universe’s kind to be O1, and T ≈ 7000◦–7500◦.

The results we have mentioned above show that the theory ofa homogeneous universe, with suppositions C, D, and E, deviatesfrom the contemporary observational data. Moreover, as it followsfrom Hubble’s work, this difference applies to the classical me-chanical theory of the Metagalaxy as well as to the relativisticcosmological theory. McVittie explained the difference, supposingthat the reason is that data (a), which, affecting the quantitativebounds significantly, are inexact. Eddington [44] explained it byarguing that other observational data are inexact. At the same time,to eliminate the difference, we actually need to take the extremenumerical values of the possible empirical data. Therefore we willexamine Suppositions C, D, and E.

To explain the difference via the falsity of Supposition E, weneed very large masses of dark intergalactic matter (which doesnot undergo any interactions that we could observe). To elimi-nate the difference we can reject Supposition D (as had beenshown in the Hubble study), however it requires the “degenerationof photons” which is hypothetical and physically unexplained.On the contrary, by the falsity of Supposition C, Shapley proposed[45, 46] his explanation of the difference, having a basis in observ-ational data.

∗The numerical values of ρ, RR

, kR2

, and Λ used by McVittie had been obtained

in [39, 40], the bounds of T had been found by him in [41, 42], where he usedan intermediate method to calculate other values. Therefore the method McVittiefollowed is logically contradictory.


§1.15 Non-uniformity of the visible part of the Universe

The Harvard studies (Shapley and others) differ from the studiesmade at the Mount Wilson observatory (Hubble and others). Al-though the former studies penetrated less into the depths of spacethan did the latter studies, they do however take more facts intoconsideration (higher percentage of galaxies in the region studied).For these reasons the Harvard studies complement the Mount Wil-son data. In points of dispute, the Harvard studies take, possibly,greater weight. Collecting the results, we make the following points:

1. Many galactic clusters of up to hundreds of galaxies exist. Thetendency for galaxies to concentrate themselves into clustersis under discussion. Hubble thinks the tendency less clearthan does Shapley. At the same time, the fact that the tend-ency itself exists is indisputable [47, 48];

2. From the galaxies which are not more distant than about3×106 parsecs, about two thirds are located in the northerngalactic hemisphere [35]. This results from the fact that themassive galactic cluster in the Virgo constellation is in thenorthern hemisphere;

3. From the galaxies which are not more distant than about3×107 parsecs, the larger part is located in the southern hemi-sphere [49, 47, 50, 46]. As shown by Shapley (see ibid.), thisresult does not contradict the Hubble data that both hemi-spheres have the same quantity of galaxies, which are notmore distant than 108 parsecs [47, 36];

4. The number of galaxies (not more distant than about 3×107

parsecs) in the strip 30◦×120◦, which covers the South Pole,being calculated for 1 square degree of the strip, has a grad-ient from the beginning of the strip to its end [50, 46];

5. The coefficient of proportionality between the red shift and thegalactic distances is different in the two hemispheres [50, 46].

The 1st and 2nd results do not contradict Supposition C. Theysuggest, exclusively, that the Universe is certainly inhomogeneousand anisotropic on a scale less than the aforementioned. The 3rdresult indicates that a density gradient in the Metagalaxy alongone of its radial directions, and also an anisotropy produced by thegradient, exist. So Supposition C is very inaccurate. The 4th resultindicates that an essential density gradient in the Metagalaxy, andalso an essential anisotropy in the observed direction, exist. So

1.15 Non-uniformity of the visible part of the Universe 43

Supposition C is severely violated. Finally, the 5th result showsthat the anisotropy of matter in the Metagalaxy exists in companywith the anisotropy of its deformations.

Taking Shapley’s results into account, the 4th result mainly,McCrea [51] set up the next problem: find theoretical correlationsbetween observable values, first, between the numerical values ofthe red shift and photometric distances, calculated with astronom-ical methods. He set up this problem within the framework of thesuppositions: (1) the General Theory of Relativity is true in theregion where the observable space objects are located; (2) some ad-ditional conditions, which do not need homogeneity and anisotropy,take a place in the region. In relation to the first supposition,McCrea took Λ=0 in the Einstein equations. An additional condi-tion he took into account was that matter can be assumed to bean ideal fluid without inner pressure. As a result of the latter, it ispossible to utilise accompanying coordinates, where

g00 = 1 ,∂g0i∂t

= 0 , (15.1)

so, besides (15.1), it would be possible to suppose time orthogonalto space everywhere, namely

g0i = 0 , i = 1, 2, 3 . (15.2)

As a reason of the first of the additional conditions, McCreapointed out that the peculiar velocities of galaxies are low. Thesecond additional condition of McCrea has no simple physical inter-pretation. The condition had been introduced by him for onlymathematical simplification∗.

Using his suppositions, McCrea found an approximate correla-tion between the numerical values of the red shift and the “project-ed distances” (which characterize distances to the observable spaceobjects), and the density of matter at the point of the observations.He pointed out that he was not successful in linking the “projecteddistances” to the distances astronomical data give, and so he wasunable to solve the problem he had set up†.

McCrea’s failure is nonrandom and has also no explanation inmathematical difficulties. It is doubtful, in general, that McCrea’s

∗An analogous declaration had been made by Friedmann [5].†“Projected distance” coincides with regular photometric distance in only the

first approximation. At the same time it is insufficient, if we want to apply thecorrelation McCrea had found.


suppositions would be sufficient for determining theoretical corre-lations between the observable values. His suppositions could besufficient for some qualitative conclusions on the evolution of theobservable fragment of the Universe. So these would be enoughfor some steps toward building the theory of an inhomogeneousuniverse. As a result, it is possible that the aforementioned corre-lations in which McCrea was interested can be obtained only in theframework of the completed theory of an inhomogeneous universe.

§1.16 The theory of an inhomogeneous universe

The initial suppositions of the theory of a homogeneous universe,namely — the supposition we have mentioned in §1.1 or Supposi-tions A and B (see §1.11), cannot be justified. Those models, whichcan be obtained under the suppositions, have the properties wehave considered in §1.12 and §1.13 to be inapplicable to the realUniverse. The supposition that the known fragment of the Universeis homogeneous and, hence, is isotropic (see Supposition C, §1.11)is in contradiction to observational data. The arbitrariness of theexisting theory and the necessity of building the relativistic theoryof an inhomogeneous universe had been understood even beforethe contradiction had been found. However, the development wasdelayed because of mathematical difficulties in the problem∗. In thisconsideration, the thesis that the Universe as a whole is identifiedwith the Metagalaxy remained valid. At the same time variousastronomers, namely — Mason, Fesenkov, Eigenson, and Krat [52,53, 54, 27, 55], conceive of the Metagalaxy spatially as limited (thatthe Metagalaxy is inhabited by numerous metagalaxies). One canlink this viewpoint, actually unnecessarily, to Lambert’s concept,according to which the Universe has a hierarchical interior of infi-nitely numerous levels [53, 54, 56].

The following suppositions for a method of building the theoryof an inhomogeneous universe are reasonable.

Suppositions All previous suppositions relate to the Universe asa whole. To replace them with the following suppositions on a

∗For instance, see [1], p. 330, 332, 363–364, 482, 486–488. Note that besides the“technical” difficulties, there is the principal difficulty that no universal boundaryconditions for infinity exist in the relativistic theory of gravitation. If the propertiesof symmetry exist in, for instance, a homogeneous universe, then the propertiescompensate the absence of the aforementioned boundary conditions. At the sametime we cannot attribute the properties to an inhomogeneous universe.

1.16 The theory of an inhomogeneous universe 45

fragment of the Universe we are considering at each stage ofthis study∗.

1. Einstein’s General Theory of Relativity is valid everywherein the fragment we are considering. In addition, althoughonly the case Λ=0 has an explicit physical sense, we canconsider the cases Λ 6=0, aiming to compare them with theformer. So we take the cosmological constant valid in theinitial equations.

2. The volume we are considering is filled with matter, consist-ing of a monotonic distribution of substance† and radiations,in general. A difficulty with this however, is that the four-dimensional energy-momentum tensor, corrected for heatfluxes, has not yet been found (see [1], p. 330). This difficultycan be passed over, if we consider the transparency of theintergalactic space within the framework of the suppositionthat radioactive transfer dominates other transfers of energy.Supposing the interaction energy of two elements of matternegligible in comparison with the energy of each element, wecan write

T νμ =(T νμ)m+(T νμ)r, (16.1)

where the first term here describes the substance free of heatfluxes, the second term describes the radiations. So we have

T = (T )m . (16.2)

3. The fragment we are considering can be spanned, step-by-step, by coordinate nets which accompany the substance. Inother words, numerous conditions, making such coordinatespossible, take a place in the given fragment of the Universe.Because the substance (galaxies) is the “skeleton” of the Uni-verse, and because we use the accompanying coordinates inobservational tests of the theory, the aforementioned coordi-nates have more physical meaning than coordinates accompa-nying the whole of the matter on the average‡. We note thatunder the suppositions we have made above, we have

(T i0)m= 0 , (16.3)

∗By studying an element of the Universe, we mean “a fragment we are consid-ering” as any infinitesimal (finite) volume, which contains this element.

†This is a “liquid”, or better, a “gas”, where galaxies are “molecules”.‡Both kinds of the accompanying coordinates are the same in a homogeneous

universe.


thus we obtain, finally

T i0 =(T i0)r. (16.4)

The way to build the theory of an inhomogeneous universe To re-tain considerations on the Universe as a whole from the be-ginning of the study. To split the study into the followingstages.

1. Consider the evolution of an element of an inhomogeneousanisotropic universe. The main point here is the followingproblem: is the evolution of universes of kind O2 really pos-sible, and what are the conditions of their evolution (if thiskind of evolution can exist)? Actually, this is the question:can the oscillating models, which do not transit through thespecial state of infinite density, exist, and what are the condi-tions? If the answer is in the negative, then the assumedidealization is not applicable to the real Universe.

The minimum of those scales where the second of the assumedsuppositions is reasonable, is of the same order as that mentionedin §1.11. To study the Universe on this scale would be most de-sirable. At the same time, we can also use a scale, larger than theminimum, only if the “elements” of the Universe we have in thescale are infinitesimal in comparison to the Metagalaxy(if the Metagalaxy is infinite, then its elements can be assumed aslarge as desirable). Taking the latter into account, results of the 1ststage of the study can be applicable to very large fragments of theUniverse.

From the 1st stage of this study, we can go to the 2nd stage or,by passing this stage, directly to the 3rd stage, where the problemof space curvature has a special meaning. For this reason, we musttake space curvature into account in the 1st stage of the study.

2. Consider the case where the Metagalaxy is spatially finite.Analogously to the previous stage, the question about applica-bility of the type O2 to all the elements of the Metagalaxy (andthe conditions of its evolution, if it is applicable) has a specialmeaning here. In other words, we ask: are the special statesof the infinite density absent throughout the four-dimensionalvolume, where the Metagalaxy is? If the answer is in thenegative, then the assumed idealization is not applicable or,probably, the Metagalaxy’s spatial boundedness cannot existeternally.

1.17 The cosmological equations: the local interpretation 47

3. Consider the case where the Metagalaxy is spatially unbound-ed and infinite (that is the Universe). The next question hasa special meaning here: are any peculiarities in this four-dimensionally infinite volume, the Universe, absent, and whatare the conditions of the peculiarities (if they exist)? To an-swer this question in the negative implies inapplicability of theaccepted idealization, or that the Lambert concept (see p. 44)is true. In connection with the aforementioned question aboutthe spatial infinitude of the Universe, the question about thespace curvature has a special meaning. Unfortunately, thecorrelation between the curvature and the topological struc-ture of space has not been studied sufficiently∗: the data wehave under consideration are apparently, exhausted withsome of the sufficient conditions of the spatial infinitude(see [57], p. 234 and 239). It is necessary to note that to applyany result we obtain in three-dimensional geometry to thecase where time is everywhere orthogonal to space, requiresfurther study.

As a matter of fact, before we study an element of an inhomo-geneous universe, we need to build a mathematical apparatus, ac-cording to the physical contents of the problem†. We need to firstfind those equations by which we will study the Universe in eachof its elements.

§1.17 The cosmological equations: the local interpretation

Let us consider the cosmological equations of a homogeneous uni-

verse we know from the foregoing. The quantities ρ, p, D=3 RR

(and

their derivatives), and also C =3 kR2

, constitute the equations thatcharacterize a homogeneous universe as a whole, as well as any ofits elements. Under such “local interpretation” of the cosmologicalequations, the quantity R does not keep its physical meaning as the“radius of the Universe”. For this reason, using (11.1) and (11.2) weexclude the radius R from the cosmological equations of gravitation(5.1) and (5.4), and also from the cosmological equation of energy

∗Except for constant curvature spaces, which play a part only in the theory of ahomogeneous universe.

†Of course, we need to develop this apparatus step-by-step at each stage ofthe study.


(4.2). As a result, the equations take the form

1

c2

(∂D

∂t+1

3D2

)

= −κ

2

(ρ+ 3

p

c2

)+ Λ , (17.1)

1

3c2D2 + C = κρ+ Λ , (17.2)

∂ρ

∂t+D

(ρ+

p

c2

)= 0 . (17.3)

The correlation between space curvature and expansions orcontractions of space, which could eliminate formula (2.2), can beobtained from (17.1–17.3). Eliminating Λ, ρ, and p from them, weobtain

∂C

∂t+2

3DC = 0 , (17.4)

then, because of (4.4) and (14.2), we have

D =1

V

∂V

∂t, (17.5)

whence∂

∂t

(3√V C

)= 0 . (17.6)

The density ρ and the pressure p describe:

(a) the state of matter throughout the space (D is the relative rateof the volume expansion of the matter);

(b) the evolution (namely — the deformations) of the substanceand its accompanying space;

(c) the curvature 13C characterizes the geometrical properties of

the accompanying space.

So having the local interpretation of the cosmological equations,we can say:

• The cosmological equations of gravitation are the correlationsthe equations of the law of gravitation set up∗ between thequantities ρ, p, D, and 1

3C (the quantities characterize thestate of matter, its deformations, and the geometrical proper-ties of the accompanying space);

∗That is, the correlations are set up by the equations of the law of gravitation,not by the equations of the law of energy, which is a consequence of the law ofgravitation.

1.17 The cosmological equations: the local interpretation 49

• The cosmological equation of energy is the correlation theequations of the law of energy set up between the aforemen-tioned quantities ρ, p, and D. In other words, this is thecorrelation between the state of the matter and deformationsof the accompanying space.

We are now going to consider the general case of the equations ofthe law of gravitation and of the law of energy in the accompanyingcoordinates.

The quantities ρ and p, characterizing the state of matter, arecontained in components of Tμν , which consist of the equation ofgravitation and the equation of energy. The quantity D, describingthe evolution (i. e. deformations) of the substance and of its ac-

companying space, must evidently be linked to the quantities ∂gik∂x0

,which are contained in the equation of gravitation and the equationof energy as well. The curvature 1

3 C, characterizing the geometricalproperties of the accompanying space, is linked to the quantities∂2gik∂xj∂xl

, which are contained in the equation of gravitation, howeverit is absent in the equation of energy. Besides those mentioned

above, some other quantities, which are linked to ∂g0α∂xβ

, can appear

in both equations (the quantities characterize the force field∗ of theaccompanying space).

Thus, in the general case, we can obtain:

• The correlations the equations of gravitation set up betweenthe quantities ρ, p, D, and 1

3C, characterizing the state ofmatter, deformations and the geometrical properties of theaccompanying space, and, possibly, the force field;

• The correlations the equations of the law of energy set upbetween the quantities ρ, p, and D, describing the state ofmatter, deformations of the space, and, possibly, the forcefield.

By analogy with the equations of a homogeneous universe, wewill refer to the first as the cosmological equations of gravitation,and to the second as the cosmological equations of energy. In gener-al, we call the correlations between the aforementioned quantities(which characterize the state of matter, deformations and the geo-metrical properties of the accompanying space, and also the forcefield) the cosmological equations. It is evident that the cosmological

∗In the sense of Classical Mechanics.


equations of an inhomogeneous universe, being a generalization ofthe cosmological equations of a homogeneous universe, permit onlytheir local interpretation.

We will use the cosmological equations for studying elements ofinhomogeneous universes.

§1.18 Friedmann’s case in an inhomogeneous universe

This study realises only the beginning of the program we havementioned above. We will limit ourselves to the first stage (see§1.16), where the main task is to consider the cosmological equa-tions and some of their consequences. From the consequences wewill consider, mainly, the evolution of the volume of an element ofan inhomogeneous universe, where we assume the matter a sub-stance∗ of positive density without any tensions or pressure. Wealso suppose the substance to be free of heat fluxes. Then, withinthe framework of arbitrary coordinates, its energy-momentum ten-sor is the tensor of an ideal fluid of infinitesimal pressure, that is

Tμν = ρdxμ

ds

dxν

ds, ρ > 0 . (18.1)

This case correlates the condition (4.7) in a homogeneous uni-verse† and, like the same case in homogeneous universe, can becalled Friedmann’s case.

To clarify that the Friedmann case in an inhomogeneous uni-verse relates to the same case in a homogeneous universe and to thegeneral case of an inhomogeneous universe, we are going to makea list of those factors which are expected in the transfer from ahomogeneous universe to an inhomogeneous one.

Factors which do not link to the presence of pressure

1. Anisotropy of deformations of elements of the accompanyingspace.

2. Inhomogeneity of their density‡.

∗That is, here Zelmanov supposed the matter a substance without radiations. —Editor’s comment. D. R.

†Compare (18.1) with formula (3.6) under the condition (4.7).‡Note that the density gradient in explicit form cannot appear in the

cosmological equations, because no derivatives of the time component of the energy-momentum tensor with respect to spatial coordinates exist in the initial relativisticequations. This can be explained from the physical viewpoint, because the term“density gradient” has no meaning with respect to a single element of the Universe.

1.18 Friedmann’s case in an inhomogeneous universe 51

3. A field of forces, which act on test-bodies only in their mo-tions with respect to the coordinates we are considering (thecoordinates which accompany to substance).

4. The correlation between the Riemannian curvature and two-dimensional directions (i. e. anisotropy of the curvature).

5. Inhomogeneity of the Riemannian curvature along all two-dimensional directions (i. e. inhomogeneity of the curvature)∗.

Factors linked to the presence of pressure

6. Viscosity, which manifests under anisotropy of the deforma-tions†.

7. Divergence of the three-dimensional stress tensor; in otherwords, inhomogeneity of pressure, linked to inhomogeneityof density.

8. A field of forces, which put the inhomogeneity of pressureinto equilibrium.

9. A heat flux, linked to the inhomogeneity of pressure.

10. A flux of momentum, linked to the heat flux.

It is evident that the facts of the first group are different to thecase we are considering from the Friedmann case in a homogeneousuniverse, the facts of the second group are different to the generalcase of an inhomogeneous universe.

The Friedmann case in the theory of a homogeneous universeis inapplicable to thermodynamics. At the same time, this case issufficient for dynamics and observational tests of the theory, andis that sole case which is applicable for comparing the theory withClassical Mechanics in strong form. Taking these as a basis forbuilding the theory of an inhomogeneous universe, we can say thatthe Friedmann case cannot be applicable to thermodynamics; it is

∗The mean curvature gradient in explicit form cannot appear in the cosmologicalequations, because it must include the third derivatives of components of the metrictensor.

†As it has been obtained in the theory of a homogeneous universe (see §1.3),matter can be assumed to be an ideal fluid. Then, from the physical viewpoint, itcan be a consequence of not only the absence of the viscosity, but also the presenceof the viscosity under isotropy of the deformations. In other words, this is thefact we have also in Classical Hydrodynamics (for instance, see [58], p. 544, underthe conditions a= b= c 6=0 and f = g=h=0). Note that the viscosity of the “fluid”,consisting of “molecules-galaxies”, is four orders greater than the viscosity of water.It can easy be calculated, taking masses, volumes, and peculiar velocities of galaxiesinto the calculation.


the sole case whereby we have any hope of finding a strong analogyto the equations of Classical Mechanics. At the same time, becausenumerous factors are linked to light pressure in an inhomogeneousuniverse, we have no knowledge of how much the Friedmann case(and its consequences) differ from the general case in the spheresof dynamics, geometry, and its observational tests. We are limitingour attention by the Friedmann case here, because mathematicalsimplicity and physical obviousness recommend to first considerinfluences of only one of the groups of the factors, namely the firstgroup of the factors, because those factors can be in action even inthe absence of any factors of the second group (the last factors ofthe aforementioned are in action under only the presence of someof the factors of the first group). In our opinion, any reasons thatpressure and its linked factors can be infinitesimal in the fragmentof the Universe we observe in the present time, have secondarymeanings.

The next case, after the Friedmann case, could be considered asan approximation to the case of an inhomogeneous universe∗. Thisis the case of

Tμν =(ρ+

p

c2

) dxμ

ds

dxν

ds−p

c2gμν , ρ > 0 , p > 0 , (18.2)

i. e. an ideal fluid — the mix of a substance and radiations underthe 7th and the 8th factors. This case corresponds to the generalcase of a homogeneous universe, and it is the general case, wherethe condition (4.6) remains unchanged.

To obtain the cosmological equations and their analysis we needto prepare the necessary mathematical apparatus and also methodsto interpret numerous mathematical results we will obtain. Thesetasks will also be included in this work.

We have given a brief survey of the theory and the tasks of ourwork. Looking forwards, we propose the following plan. To preparethe necessary mathematical apparatus (Chapter 2); To apply the ap-paratus to equations of physics (Chapter 3); To use the equations toobtain the cosmological equations and their consequences, consid-ering the Friedmann case in an inhomogeneous universe (Chap. 4).

Keeping in mind that programme we have set forth in §1.16, wewill complete each of the Chapters with results, more than it wouldbe necessary.

∗It has been partially developed by the author, but the results are not includedin this work.

1.19 The mathematical methods 53

§1.19 The mathematical methods

Considering the proposed contents of Chapter 2, we must first con-sider the following problem: is the use of some special coordinateframes of the systems accompanying the Universe’s substance, pos-sible and appropriate?

Most simple are those coordinates where

g0α = δ0α . (19.1)

Very simple and physically rational are the coordinates where

∂

∂xβ(gαβ

√−g)= 0 , (19.2)

proposed first by Lanczos [59]. Such coordinates have been used inphysical studies by Fock (the “harmonic coordinates” [60]).

However, generally speaking, both coordinate frames of theabove are not of the accompanying coordinates kind∗. So, consider-ing the conditions (19.1) and (19.2), we can associate the accom-paniment condition with only the case α=0, where the choiceof time coordinate is fixed. So, using the special coordinates (ofthe accompanying ones) in general equations cannot give muchsimplification. Moreover, this is not good mathematically, because acircle of those simplifications, which could be possible in the partialcases, is much narrower under the special coordinates. Finally,this is inappropriate, for the following physical reasons. The specialcoordinate frames obstruct selection of physical quantities which:

(1) are invariant with respect to those transformations, wherethe accompaniment of the coordinates to the substance is notviolated;

(2) contain not only the transformations of the spatial coordinates

xi′= xi

′(x1, x2, x3) , i = 1, 2, 3 , (19.3)

but also any transformation of the time coordinate, i. e.

xα′ = xα′(x0, x1, x2, x3) , α = 0, 1, 2, 3 . (19.4)

Because of the physical equivalence of all time coordinates,among the things measured in the body of reference that we use,

∗We will use the harmonic coordinates, moving with respect to theaccompanying ones, in §4.26.


the invariance with respect to transformations (19.4), in otherwords, the property of “chronometric invariance”, must be a prop-erty of all the main quantities and equations which characterize thisbody of reference. For this reason we retain attempts to apply anyspecial coordinates, and so we set forth the main task of Chapter 2as the introduction of a three-dimensional tensor calculus, the mainquantities and operators of which have this additional property ofchronometric invariance with respect to transformations (19.4). Wewill refer to this new mathematical apparatus as the chronometric-ally invariant tensor calculus. Accordingly, we will refer to itsobjects and operators as chronometrically invariant tensors∗, whichcharacterize deformations of the accompanying space, its curvature,and its metric.

Chapter 2 will be arranged into numerous thematic sections.

A. §2.1–§2.4 introduce three-dimensional quantities, both invari-ant with respect to (19.4) and non-invariant. The non-invariantquantities will be very helpful because of their property totake, as a result of our choice of a specific coordinate of time,the necessary numerical values†.

B. §2.5–§2.7 introduce the generalized operators of differentia-tion with respect to time and spatial coordinates, invariantwith respect to (19.4). The sections also introduce quantities(“force values” — the vector Fi and the antisymmetric tensorAjk) which characterize non-commutativity of the operators‡.

C. §2.8–§2.12 introduce the chr.inv.-metric tensor, the chr.inv.-tensor of the space deformations, the correlation between thetensors, and also consequences of this correlation. Equation(11.22), giving the correlation, is the most important result ofthe entire study§.

∗Chr.inv.-tensors, in brief. Initially, Zelmanov called chronometric invarianttensors in-tensors, and the whole mathematical apparatus the in-tensor calculus.However the terms were unsuccessful, so Zelmanov replaced them with the morereasonable terms chronometrically invariant tensors and the chronometricallyinvariant calculus. — Editor’s comment. D. R.

†See the method to vary the potentials — §3.7, §3.12, §3.17, and §3.20.‡§2.7 gives the necessary and sufficient conditions of g00=1 and

∂g0i∂t

=0 (15.1),and also g0i=0 (15.2) can be written as Fi≡ 0 and Ajk ≡ 0, respectively. Theoperators give a possibility of bringing two cases, where the world-lines of an idealfluid’s current are Eisenhart [61] geodesics, together into a single case, which is de-

fined by the condition∗∂p∂xi

≡ 0 in the accompanying coordinates (p is pressure).§Under accompanying coordinates (not only in relativistic mechanics but also in

1.20 Relativistic physical equations 55

D. §2.13–§2.15 introduce generalized covariant differentiation inself-deforming spaces, invariant with respect to (19.4). Thesections also give the necessary generalizations of theRiemann-Christoffel tensor and Einstein’s tensor.

E. §2.16–§2.19 introduce chr.inv.-tensor quantities, which de-scribe rotational motions. The correlation between relativerotations of the elements of the self-deforming accompanyingspace and the space deformations is given there. This corre-lation, as seen from formula (19.19) in §2.19, implies a resultsimilar to the well-known equation

rot rot ~v = grad div ~v −∇2~v (19.5)

if: (1) we re-write the equation in the form

1

2rot rot ~v =

(grad div ~v −

1

2rot rot ~v

)−∇2~v , (19.6)

and (2) we take ~v to be the vector of velocity.

F. §2.20–§2.22 introduce chr.inv.-tensor quantities which char-acterize the curvature of the self-deforming accompanyingspace. The quantities are the chr.inv.-tensor generalizations ofthe curvature tensors, the curvature scalar, and Ricci’s tensor.

Using only the relativistic equation of the four-dimensional in-terval ds2 as a basis for this study, Chapter 2 does not use anyadditional equations or additional physical requirements.

§1.20 Relativistic physical equations

Because we will deduce the cosmological equations in Chapter 4, wemust consider relativistic physical equations before that Chapter.This will be done in Chapter 3. We must first clarify the physical(mechanical) sense of the “power quantities” Fi and Ajk on theone hand, and, on the other hand, obtain chr.inv.-tensor quantitieswhich describe force fields. Both tasks need to establish the equa-tions of geodesic world-lines in chr.inv.-tensor form and to comparethem with the equations of Classical Mechanics. We must introducechr.inv.-tensor quantities which characterize states of matter. Thistask requires deduction of components of the energy-momentum

Classical Mechanics), this correlation implies that a third interpretation of motionsof continuous media exists, besides those of Euler and Lagrange.


world-tensor in chr.inv.-tensor form. Finally, introducing chr.inv.-tensor quantities which characterize states of matter, deformationsof space, space curvature, and forces acting in the space, we mustobtain equations of the law of gravitation and the law of energy inchr.inv.-tensor form. So the main task of Chapter 3 is to translatethe main relativistic tensor quantities and the equations into the“specific language” of the mathematical apparatus of chronometricinvariants (see the apparatus itself in Chapter 2), clarifying the me-chanical sense of the “power quantities” simultaneously.

Let us make a list of the main themes contained in Chapter 3.

A. §3.1–§3.3 give three-dimensional tensor equations for compo-nents of the world-metric tensor and also for Christoffel’sworld-symbols of the 1st and the 2nd kind. Very significant inthe consequences (see §3.14–§3.17) is the fact that the afore-mentioned tensor world-quantities Q, all indices of which arenot zero, are linked to the appropriate three-dimensional ten-sor quantities T of the same indices by the linear equations

Q = ±T + a , (20.1)

where the additional term a can be a three-dimensional quan-tity or zero.

B. §3.4–§3.6 introduce (after the chr.inv.-vector of the velocityof light introduced in §2.9) chr.inv.-tensor characteristic ofa point-mass — its mass, energy, and momentum, invariantwith respect to (19.4). The generalized total derivatives of thequantities with respect to time are also given there. As we ex-pect, after we apply the chr.inv.-tensor apparatus, these rela-tions between the physical quantities considered are similar tothe relations we know from the Special Theory of Relativity.

C. §3.7–§3.9 deduce the equations of world-geodesics in chr.inv.-tensor form.The chr.inv.-equations obtained give the dynamicequations and the theorem of energy of a point-mass, whichcan be compared with the appropriate equations of ClassicalMechanics. This comparison justifies the terminology we haveused for “power quantities”, because they are characteristicsof force fields; namely, the chr.inv.-vector Fi plays the partof gravitational inertial force, calculated with a unit mass, thechr.inv.-tensor Ajk (or the chr.inv.-vector Ωi, appropriatingthe Ajk) plays the part of the momentary angular velocity ofthe absolute rotation of the reference frame in the formula

1.20 Relativistic physical equations 57

for the Coriolis force. As it is easy to see that the “powerquantities” Fi and Ωi relate those factors (see §1.18), whichcan be expected in the transfer from a homogeneous universeto an inhomogeneous one: Fi correlates the 8th factor (thefield of forces, which put the effect of the spatial inhomogene-ity into equilibrium), Ωi correlates the 3rd factor (the field offorces, which act only upon bodies in motion with respect tothe coordinates accompanying the substance). In our opinion,this result is very important∗.

D. §3.10–§3.13 obtain components of the energy-momentumworld-tensor in chr.inv.-form, express them with the massdensity, the momentum density, and the kinematic stress ten-sor, invariant with respect to (19.4). Using the results, weobtain world-equations of the law of energy in chr.inv.-tensorform. The equations obtained for the law of energy (the scalarequation for the energy density, the vector equations for themomentum density) contain only chr.inv.-tensor quantities,which characterize states of matter, its evolution, the spacedeformations, and forces acting in the space.

E. §3.14–§3.17 give components of Einstein’s world-tensor inchr.inv.-tensor form. This operation requires tedious formalalgebra, but results in numerous very compact formulae. As aresult of §3.1–§3.3, spatial components of the Einstein world-tensor are linked to components of the appropriate three-dimensional tensor by equations like (20.1).

F. §3.18–§3.24 develop the equations of Einstein’s law of grav-itation into chr.inv.-tensor form. As a result, we obtain thescalar equation of gravitation, the vector equation, and thetensor equation. The equations consist of only chr.inv.-tensorquantities, which characterize states of matter, its evolution,the space deformations, and forces acting in the space. Itis interesting that the vector equation can be considered asa differential equation with respect to the angular velocityvector of the “geodesic precession” (see §4.10).

∗In particular, this result has the following consequences. The physical senseof the supposition (see §1.15) that time is everywhere orthogonal to space g0i=0(15.2), which was unknown to Friedmann and McCrea, is the absence of the Corioliseffect (the absence of the “dynamic absolute rotation”, in the terminology of the

Chapter 4). The physical sense of the conditions g00=1,∂g0i∂t

=0 (15.1) is merely

that those forces which could put the pressure gradient into equilibrium, are absent.


All the results of Chapter 3 are not linked to any limiting suppo-sitions, related to kinds of matter or relative motions of the coordi-nate systems we use. The entire problem statement and the resultsare independent of both the mathematical viewpoint and the phys-ical viewpoint. This fact, in our opinion, suggests the following. Themathematical apparatus of chronometric invariants we give in Chap-ter 2 could be a natural form of physical equations in those cases,where the complex of coordinate systems, which are at rest with re-spect to each other, plays a special part. One such case is relativisticcosmology, because it uses accompanying coordinate frames.

§1.21 Numerous cosmological consequences

Beginning with Chapter 4, which is actually the cosmological partof this study, we will give numerous notes on the cosmologicalequations in the Friedmann case, namely — on those 10 factors(see §1.18), which differentiate the inhomogeneous universe fromthe homogeneous one.

The absence of pressure and also of the 6th, 7th, 9th, and the10th factors, linked to it, must be taken into account. It is necessaryto do this when we introduce quantities which characterize states ofmatter and its evolution (if the quantities are taken in coordinatesaccompanying the substance), into the cosmological equations. Theabsence of the 8th factor, i. e. when the vector Fi is zero, must beobtained as a consequence of the cosmological equations. Lookingat the factors of inhomogeneity, we can say that the 2nd and the 5thfactors cannot appear in explicit form in the cosmological equations.For this reason the factors manifest their links to the other factorsof inhomogeneity (the 1st, 3rd, and the 4th factors) only underconsideration of a finite or infinite volume of the space. The re-maining factors of inhomogeneity (the 3rd of them appears as thevector Ωi) are linked to the nonequivalence of two-dimensionaldirections∗, so they can be called the factors of anisotropy. Thesefactors can affect the evolution of “isotropic” characteristics of anyelement of the Universe — its volume, density, mass, and the meancurvature of the space, linked to the factors and also one to anotherby the cosmological equations. From this we can appreciate that,concerning the evolution of one of the “isotropic” characteristics,we need to consider not only the evolution of other “isotropic” char-

∗The nonequivalence of two-dimensional directions in three-dimensional spaceis the same as the nonequivalence of one-dimensional directions.

1.21 Numerous cosmological consequences 59

acteristics (just as we did in the theory of homogeneous universe)but also the factors of anisotropy. The factors of inhomogeneity canbe expressed as relations between themselves on the one hand, andthe relations between them and the evolution of the “isotropic”characteristics on the other hand. To do this we must take thecosmological equations and replace the tensor equation we are writ-ing in the abstract form

Bki = 0 (21.1)

with the equivalent system of equations: (1) the scalar equation

B = 0 (21.2)

which we obtain by contraction of equation (21.1); and (2) the ten-sor equation

Bki −1

3hkiB = 0 , (21.3)

which becomes an identity after contraction (see the “main form”of the cosmological equations).

After the foregoing has been established, we formulate the maintask of our cosmological problem (see the beginning of §1.18) moreprecisely. So the main task of Chapter 4 is to obtain the cosmolo-gical equations for the Friedmann case in an inhomogeneous uni-verse, and also numerous their consequences thereof, which char-acterize: (1) factors, which are unnatural in homogeneous models;(2) effects of the unnatural factors on the evolution of the main“isotropic” characteristics of an arbitrary element of an inhomo-geneous universe (the evolution of its volume being the main task).

Chapter 4 will also be arranged into numerous thematic sections.

A. §4.1–§4.4 introduce suppositions on those kinds of matterwhich pertain to the Friedmann case, and also coordinatesaccompanying the substance. To describe deformations andrelative rotations of elements of the substance, we use quan-tities characterizing the analogous motions of the elementsof the space. We also introduce the possibility of using theaccompanying coordinates, on a basis of general suppositionson motions of the substance.

B. §4.5–§4.8 give the cosmological equations in their ”mainform”, and also set forth the possibility of introducing theprimary coordinate of time at any given point of the space(this possibility is derived from the fact that the vector Fi


can be reduced to zero at any point). The “main form” of thecosmological equations shows that if the scalar cosmologicalequations characterize an element of the Universe, independ-ent of spatial directions, and the tensor equations characterizeanisotropy of the element, then the vector cosmological equa-tion of gravitation links the evolution of the element to theevolution of its neighbouring elements.

C. §4.9–§4.13 consider the factors of anisotropy and also rela-tions between anisotropy and inhomogeneity. Let us mentionof those results that: (1) the vector cosmological equation, inits mechanical sense, is a differential equation with respectto the angular velocity vector of the “geodesic precession”;(2) dynamic absolute rotations cannot disappear or appear,which is a peculiarity of this factor of anisotropy; (3) thelaw of transformations of the vector Ωi under deformationsof the element; (4) the deformations can be anisotropic in theabsence of other factors of anisotropy, which is a peculiarityof only the anisotropy of deformations; (5) the presence ofinhomogeneities in a finite volume is necessary, if the volumeis isotropic; (6) a static model, which differs from the Einsteinstatic model, can manifest when Ωi≡ 0.

D. §4.14–§4.18 consider the scalar cosmological equations, gene-ralizing the equations of the theory of a homogeneous uni-verse. Using the scalar equations, we obtain the laws of trans-formations of the mean curvature of the space and of thedensity of the substance under transformations of the volumeof the element. We also make some preparations in considera-tion of possible kinds of evolution of the volume under trans-formations of an arbitrary time coordinate. Let us mentionof those results that: (1) anisotropy of the space curvaturedoes not act directly on the evolution of the “isotropic” char-acteristic of the element; (2) the mass and energy of theelement remain unchanged (as for a homogeneous universewhen ρ=0); (3) the mean curvature of the space can changesign, if the anisotropy factors are present in the space; (4) themean curvature (taken in an element) undergoes changes dueto the space deformations, just as in homogeneous universe,if the element of an inhomogeneous universe is isotropic.

E. §4.19–§4.23, as well as §1.7, consider states between whichmonotonic transformations of the volume of an arbitrary ele-

1.21 Numerous cosmological consequences 61

ment are conceivable. For an isotropic element, we obtain thesame results as in a homogeneous universe. The anisotropyfactors lead to: (1) additional different states, which are pos-sible for given numerical values of the cosmological constantand the mean curvature of the space; (2) states which are un-known in the theory of a homogeneous universe; (3) removalof the prohibition of the transformations, limited from aboveand below, which is most important.

F. §4.24–§4.28 consider possible kinds of evolution of the volumeof an arbitrary element. First, we consider the evolution inthe interval of monotonic deformations. Then, in that interval,where the volume and its density remain unchanged. The an-isotropy factors make the types of evolution which are possiblefor given numerical values of the cosmological constant, morenumerous. The aforementioned factors result new kinds ofevolution, which are impossible in the case of a homogeneousuniverse. In particular, it is very essential to note that theanisotropy factors result in a new oscillating model of theO2 kind, which does not transit through the special state ofinfinite density. Let us mention the most important results:(1) for an element of an inhomogeneous isotropic universe,the possible kinds of transformations of its volume whichare possible for given numerical values of the cosmologicalconstant and the given sign of the mean curvature (or, whenthe mean curvature is zero), are the same as those in a homo-geneous universe; (2) for anisotropic deformations in the ab-sence of dynamic absolute rotations, new kinds of evolution(in comparison with a homogeneous universe) can exist onlyfor Λ> 0,∗ and the kind O2 can exist only when the meancurvature is positive; (3) in the general case of anisotropy,including also dynamic absolute motions, new kinds of evolu-tion (between which is the kind O2) are possible for anynumerical values of the cosmological constant, so that thereare no limitations on the sign of the mean curvature.

The results show that dynamic absolute rotations (the Corioliseffect) can play a very important part in the Universe.

♦

∗For this reason, only the kinds M1 and O1 are possible under McCrea’s suppo-sitions (see §1.15).

Chapter 2

THE MATHEMATICAL METHODS

§2.1 Reference frames

The general transformations of coordinates of a system S into co-ordinates of any other system S′

xα′ = xα′(x0, x1, x2, x3) , α = 0, 1, 2, 3 (1.1)

can be written in the form

x0′= x0

′(x0, x1, x2, x3)

xi′= xi

′(x0, x1, x2, x3) , i = 1, 2, 3

}

. (1.2)

Greek indices can be 0, 1, 2, 3, Latin indices only 1, 2, 3.∗ Let usconsider a particular case of the transformations (1.2), where thecondition

∂xi′

∂x0≡ 0 (1.3)

holds, i. e. transformations like

x0′= x0

′(x0, x1, x2, x3)

xi′= xi

′(x1, x2, x3)

}

. (1.4)

Because of (1.3), we have

dxi′=∂xi

′

∂xjdxj (1.5)

and alsodxj = 0 , (1.6)

∗In other words, Zelmanov denotes space-time indices in Greek, and spatialindices Latin. This implies that he considers a four-dimensional space-time ofsignature (+−−−), where time is real and spatial coordinates are imaginary. Becauseof the latter fact, the three-dimensional observable impulse (the projection of thefour-dimensional impulse vector on the observer’s spatial section) is positive, forbenefit in practical calculations. — Editor’s comment. D. R.

2.1 Reference frames 63

so we obtaindxi

′= 0 . (1.7)

On the other hand

dxj =∂xj

∂x0′dx0

′+∂xj

∂xi′ dx

i′. (1.8)

Because equation (1.6) gives (1.7), we have

∂xj

∂x0′≡ 0 . (1.9)

We distinguish the term “reference frame” from the term “co-ordinate frame”. We will say that coordinate frames S and S′ areof the same reference frame, if the conditions (1.3) and (1.9) aretrue for them. Besides this, the systems S and S′ are of the samereference frame, because

∂xk′′

∂x0′≡ 0 . (1.10)

Since in the general case we have

∂xk′′

∂x0=∂xk

′′

∂x0′∂x0

′

∂x0+∂xk

′′

∂xj′∂xj

′

∂x0, (1.11)

(1.3) and (1.10) give∂xk

′′

∂x0≡ 0 , (1.12)

i. e. the coordinate frames S and S′′ are of the same referenceframe. In other words, if the coordinate frames S and S′ are linkedby the transformations (1.4) and the coordinate frames S and S′′

are linked by the same transformations (1.4), then the systems Sand S′′ are also linked by the transformations (1.4).

In this way we can define a reference frame as the sum of thosecoordinate systems which are linked to one another by the trans-formations (1.4). From the viewpoint of mechanics, we can specifya physical sense of a reference frame in general, via equation (1.5):

A reference frame is a complex of those coordinate frameswhich are at rest with respect to one another.

Let us take any reference frame and operate only with transforma-tions (1.4).We can modify them with the system of transformations:

64 Chapter 2 The Mathematical Methods

(1) where the spatial coordinates change but the time coordinateremains unchanged

x0′= x0

′

xi′= xi

′(x1, x2, x3)

}

, (1.13)

and (2) where the time coordinate changes but the spatial coordi-nates remain unchanged

x0′= x0

′(x0, x1, x2, x3)

xi′= xi

′

}

. (1.14)

Let us suppose that we have a relation which is true in this ref-erence frame. For its form to remain unchanged under any point-transformations of the four coordinates inside the reference frameit is a necessary and sufficient condition that the relation mustretain its form, first, under the transformations (1.13) and second,under the transformations (1.14). So the relation must retain itsform under the transformations

xi′= xi

′(x1, x2, x3) , (1.15)

x0′= x0

′(x0, x1, x2, x3) . (1.16)

§2.2 An observer’s reference space. Sub-tensors

We will use the three-dimensional space of a reference frame, thereference space in brief, defining any point of this space by a world-line

xi = ai , i = 1, 2, 3 , (2.1)

in this reference frame (ai are some numbers). Then the transfor-mations (1.15) can be considered as coordinate transformations inthis space, so there is the possibility of using three-dimensionaltensor calculus here. We call tensors of this three-dimensional cal-culus sub-tensors to distinguish them from the usual four-dimen-sional tensors. It is evident that the world-invariants and time com-ponents of world-tensors are sub-invariants (the three-dimensionalinvariants), while space-time components and spatial componentsof any world-tensor consist of sub-tensors, the ranks of which arethe numbers of their respective significant (non-zero) indices. As amatter of fact, the subscripts imply covariance and the superscripts

2.2 An observer’s reference space. Sub-tensors 65

contravariance. In general, we can write the symbolic equality

(1 + t)r= t0 + rt1 +

r (r − 1)1×2

t2 +r (r − 1) (r − 2)

1×2×3t3 + . . . , (2.2)

which implies the following: any (1+ t)-dimensional tensor of rankr splits into t-dimensional tensors, namely — one t-dimensionaltensor of zero rank (the invariant), r tenors of the 1st rank (the

vectors),r(r− 1)1×2

tensors of the 2nd rank,r(r− 1)(r− 2)

1×2×3tensors of

the 3rd rank, etc. (t=3 in this case). It is to be understood thatif the initial (1+ t)-dimensional tensor is symmetric, then some ofthe t-dimensional tensors of the initial tensor can be the same. Itis easy to see, for instance, that the covariant metric world-tensorgμν splits into the sub-invariant g00, the covariant sub-vector g0i,and the covariant symmetric sub-tensor gik of the 2nd rank. Thecontravariant metric world-tensor gμν splits into the sub-invariantg00, the contravariant sub-vector g0i, and the contravariant sym-metric sub-tensor of the 2nd rank gik. Naturally, the transforma-tions (1.13) give

g ′00 = g00 , g ′0i = g0j∂xj

∂xi′ , g ′ik = gjl

∂xj

∂xi′∂xl

∂xk′ , (2.3)

g00′= g00 , g0i

′= g0j

∂xi′

∂xj, gik

′= gjl

∂xi′

∂xj∂xk

′

∂xl. (2.4)

Numerous Christoffel world-symbols of the 1st and the 2ndrank are also sub-tensor quantities. Really, considering the generaltransformations of the Christoffel symbols∗

Γ ′μν,σ = Γαβ, ξ∂xα

∂xμ′∂xβ

∂xν ′∂xξ

∂xσ ′+ gεξ

∂2xε

∂xμ′xν ′∂xξ

∂xσ ′, (2.5)

Γσ ′μν = Γξαβ

∂xα

∂xμ′∂xβ

∂xν ′∂xσ ′

∂xξ+

∂2xξ

∂xμ′xν ′∂xσ ′

∂xξ(2.6)

and taking (1.13) into account, we obtain

Γ ′00,0 = Γ00,0 , Γ ′00,i = Γ00,j∂xj

∂xi′ ,

Γ ′0i,0 = Γ0j,0∂xj

∂xi′ , Γ ′0i,k = Γ0j,l

∂xj

∂xi′∂xl

∂xk′

, (2.7)

∗Compare formula (2.6) with (33) of [62], p. 412. Formula (2.5) can be obtainedfrom (2.6) without problems.


Γ0 ′00 = Γ

000 , Γi ′00 = Γ

j00

∂xi′

∂xj, Γ0 ′

0i = Γ00j

∂xj

∂xi′ ,

Γk ′0i = Γ

l0j

∂xj

∂xi′∂xk

′

∂xl, Γ0 ′

ik = Γ0jl∂xj

∂xi′∂xl

∂xk′

. (2.8)

In other words, we have: Γ00,0 and Γ000 are sub-invariants; Γ00,i ,Γ0i,0, and Γ00i are covariant sub-vectors; Γi00 is a contravariant sub-vector; Γ0i,k and Γk0i are mixed symmetric covariant sub-tensors ofthe 2nd rank, Γ0ik is a symmetric covariant sub-tensor of the 2ndrank. At the same time, Γik,0 , Γik,j , and Γjik are not sub-tensors.

We will introduce new sub-invariants, sub-vectors, and sub-tensors. For instance, the sub-invariant w and the sub-vector vi,which are given by the equations

g00 =(1−

w

c2

)2, (2.9)

g0i = −(1−

w

c2

) vic, (2.10)

1−w

c2> 0 . (2.11)

§2.3 Time, co-quantities, and chr.inv.-quantities

We are going to consider the transformations of time (1.16). We willrefer to sub-invariants, sub-vectors, and sub-tensors as well as anyother quantities, which change their form under the transforma-tions (1.16), as co-quantities, that is, co-invariants, co-vectors, andco-tensors∗. Sub-invariants, sub-vectors, and sub-tensors, invariantwith respect to the transformations of time (1.16), will be referredto as chronometrically invariant quantities (in brief — chronometricinvariants) thus, chr.inv.-invariants, chr.inv.-vectors, and chr.inv.-tensors. It is easy to see that g00, g0i, gik, g00, and g0i are co-quantities, while gik is chr.inv.-tensor.

THEOREM† We assume that Aij...k00...0 is the component of a world-tensor, all superscripts of which are significant, while all m sub-scripts are zero. Next, we assume that B00...0 is the time component

∗As you can see, the prefix “co” here and below has a different sense to that inWeyl’s geometry (for instance, see [7], p. 380).

†We call this theorem Zelmanov’s theorem. — Editor’s comment. D. R.

2.4 The potentials 67

of a covariant world-tensor of n-th rank. Then, because of (1.16)or (1.14), we have

Aij...k ′00...0 = A

ij...k00...0

(∂x0

∂x0′

)m, (3.1)

B ′00...0 = B00...0

(∂x0

∂x0′

)n, (3.2)

and so the quantity

Qij...k =Aij...k00...0

(B00...0)mn

(3.3)

is the component of a contravariant chr.inv.-tensor. We will use g00for B00...0, i. e.

Qij...k =Aij...k00...0

(g00)m2

. (3.4)

It should be noted,Γi00g00 is chr.inv.-tensor quantity (namely —

the chr.inv.-vector), as it easy to see.

§2.4 The potentials

We will refer to the co-invariant w and the co-vector vi as the scalarpotential and the vector potential, respectively.

Let us show, taking any reference frame, that we can transformthe time coordinate in that way, where the quantities w, v1, v2, v3,for any pre-assigned world-point∗

xσ = aσ , aσ = constσ , σ = 0, 1, 2, 3, (4.1)

can take any preassigned system of their numerical values (w)a,(v1)a, (v2)a, (v3)a the condition (2.11) permits. Then, assuming that

x0 = Aσxσ , Aσ = constσ , (4.2)

we obtain

(g00)a = (g00)a (A0)2 , (g0i)a =

[(g00)aAi + (g0i)a

]A0 , (4.3)

A0 =

√(g00)a√(g00)a

, Ai =1

√(g00)a

[√(g0i)a√(g00)a

−

√(g0i)a√(g00)a

]

, (4.4)

∗We denote the transformed quantities by a tilde, while the symbol a marks thesame quantities at the world-point (4.1).


or, in the alternative form,

A0 =1

1−(w)ac2

[

1−(w)a

c2

]

, Ai =(vi)a − (vi)a

c

[

1−(w)ac2

] . (4.5)

So the univalent numbers Aσ can be found independently ofthe numerical values of the potentials at the world-point under theprevious time coordinate and their possible numerical values underany new time coordinate.

We will refer to this method as the method of variation ofpotentials. As a matter of fact, this method provides a means bywhich the potentials any necessary numerical values at any world-point we are considering. Later in this study we will use thismethod of variation of potentials many times. Of particular im-portance will be the case where we make the potentials zero at apre-assigned world-point.

§2.5 Chr.inv.-differentiation

The standard operators for differentiation with respect to the timecoordinate and spatial coordinates, namely

∂

∂x0,

∂

∂xi, (5.1)

are, generally speaking, non-invariant with respect of the transfor-mations (1.14). In fact, we have

∂

∂x0′=

∂

∂x0∂x0

∂x0′6=

∂

∂x0, (5.2)

∂

∂xi′ =

∂

∂xi+

∂

∂x0∂x0

∂xi′ 6=

∂

∂xi. (5.3)

So we will refer to the usual differentiation as co-differentiation.Besides this, we can introduce a generalization of the usual dif-ferentiation, namely — chronometrically invariant differentiationwith respect to time and spatial coordinates, operators of whichare invariant with respect of the transformations (1.14).

We assume that an arbitrary frame of coordinates x0, x1, x2, x3

exists. Let us introduce a new system of coordinates x0, x1, x2, x3,which is of the same reference frame. Their differences are onlythat this new (tilde) coordinate frame has: (1) another time co-

2.5 Chr.inv.-differentiation 69

ordinatex0 = x0(x0, x1, x2, x3)

xi = xi

}

(5.4)

and (2) zero potentials at the world-point (4.1) so that

(g00)a = 1 , (g0i)a = 0 . (5.5)

Hence we have

(g00)a

(∂x0

∂x0

)2

a

= 1 , (g00)a

(∂x0

∂xi

)

a

+ (g0i)a = 0 . (5.6)

Considering the world-point (4.1), let us take the derivativeswith respect to time and spatial coordinates of the new system atthe given point. Next, let us transform the new coordinates into theold coordinate frame. As a result, we obtain

(∂

∂x0

)

a

=

(∂

∂x0

)

a

(∂x0

∂x0

)

a

, (5.7)

(∂

∂xi

)

a

=

(∂

∂xi

)

a

+

(∂

∂x0

)

a

(∂x0

∂xi

)

a

. (5.8)

At the same time, formula (5.6) gives

(∂x0

∂x0

)

a

=1

√(g00)a

,

(∂x0

∂xi

)

a

= −(g0i)a(g00)a

, (5.9)

therefore (∂

∂x0

)

a

=1

√(g00)a

(∂

∂x0

)

a

, (5.10)

(∂

∂xi

)

a

=

(∂

∂xi

)

a

−(g0i)a(g00)a

(∂

∂x0

)

a

. (5.11)

Because (5.10) and (5.11) are true at any world-point in anysystem of coordinates x0, x1, x2, x3, the differential operators

∗∂

∂x0≡

1√g00

∂

∂x0, (5.12)

∗∂

∂xi≡

∂

∂xi−g0ig00

∂

∂x0(5.13)


must be invariant with respect to the transformations (1.14) as well.We can see this fact directly. Naturally, we have

1√g00

∂

∂x0=

1√

g′00

(∂x0

′

∂x0

)2∂

∂x0′∂x0

′

∂x0=

1√g′00

∂

∂x0′, (5.14)

∂

∂xi−g0ig00

∂

∂x0=

∂

∂xi−

g0i√g00

1√g00

∂

∂x0=

∂

∂xi′+

∂

∂x0′∂x0

′

∂xi−

−∂x0

′

∂x0

(

g′00∂x0

′

∂x0+ g′0i

)

√

g′00

(∂x0

′

∂x0

)21

√g′00

∂

∂x0′=

∂

∂xi′ −

g′0ig′00

∂

∂x0′.

(5.15)

For this reason we accept the asterisk-marked operators (5.12)and (5.13) the chr.inv.-differential operators with respect to timeand spatial coordinates, respectively. The method we employedfor obtaining the operators, namely — formulae (5.10) and (5.11),shows the geometrical sense which chr.inv.-differentiation has inthe four-dimensional world. Chr.inv.-differentiation with respectto the time coordinate is the same as differentiation with respectto the local time of an observer at the point of his observations.Chr.inv.-differentiation with respect to spatial coordinates is spatialdifferentiation along a curve, which is orthogonal to the time lineof the observer’s reference frame. In fact, we have

(d

ds

)

xi=const

=∂

∂xα

(dxα

ds

)

xi=const

=∂

∂x0

(dx0

ds

)

xi=const

=1

√g00

∂

∂x0, (5.16)

( ∗∂

∂xi

)

g0i=0

=∂

∂xi. (5.17)

Let us give other formulae expressing the operators of chr.inv.-differentiation. It is evident that

∗∂

∂x0=

c2

c2 − w∂

∂x0, (5.18)

∗∂

∂xi=

∂

∂xi+

c vic2 − w

∂

∂x0. (5.19)

2.6 Changing the order of chr.inv.-differentiation 71

If we introducev0 =

w

c, (5.20)

then we have∗∂

∂xσ=

∂

∂xσ+

vσc− v0

∂

∂x0. (5.21)

If we introduce

t =x0

c, (5.22)

then we have∗∂

∂t=

c2

c2 − w∂

∂t, (5.23)

∗∂

∂xi=

∂

∂xi+

vic2 − w

∂

∂t. (5.24)

In addition to the above, it should be noted that if we assumethat Q is a chr.inv.-quantity, i. e.

Q′ = Q , (5.25)

the quantity satisfies the transformations (1.14), i .e.

∗∂Q′

∂xσ ′=

∗∂Q

∂xσ, (5.26)

so the chr.inv.-derivative of any chr.inv.-quantity is also a chr.inv.-quantity.

It easy to see that the chr.inv.-derivative of a chr.inv.-tensorwith respect to a time coordinate is a chr.inv.-tensor of the samerank. The chr.inv.-derivative of a chr.inv.-invariant with respect tospatial coordinates is a covariant chr.inv.-vector (chr.inv.-differen-tiation of non-zero rank tensors with respect to spatial coordinateswill be considered below).

§2.6 Changing the order of chr.inv.-differentiation

Let us suppose that quantities we are differentiating satisfy theconditions

∂2

∂xi∂t=

∂2

∂t ∂xi, (6.1)

∂2

∂xi∂xk=

∂2

∂xk∂xi. (6.2)


Then we have, respectively, for the first of the conditions

∗∂2

∂xi∂t−

∗∂2

∂t ∂xi=

∗∂

∂xi

( ∗∂

∂t

)

−∗∂

∂t

( ∗∂

∂xi

)

=

=∂

∂xi

(c2

c2 − w∂

∂t

)

+vi

c2 − w∂

∂t

(c2

c2 − w∂

∂t

)

−

−c2

c2 − w∂

∂t

(∂

∂xi

)

−c2

c2 − w∂

∂t

(vi

c2 − w∂

∂t

)

=

=c2

(c2 − w)2∂w

∂xi∂

∂t+

c2

c2 − w∂2

∂xi∂t+

c2vi(c2 − w)3

∂w

∂t

∂

∂t+

+c2vi

(c2 − w)2∂2

∂t2−

c2

c2 − w∂2

∂t ∂xi−

c2

(c2 − w)2∂vi∂t

∂

∂t−

−c2vi

(c2 − w)3∂w

∂t

∂

∂t−

c2vi(c2 − w)2

∂2

∂t2=

=c2

(c2 − w)2

(∂w

∂xi−∂vi∂t

)∂

∂t=

1

c2 − w

(∂w

∂xi−∂vi∂t

) ∗∂

∂t,

(6.3)

and for the second

∗∂2

∂xi∂xk−

∗∂2

∂xk∂xi=

∗∂

∂xi

( ∗∂

∂xk

)

−∗∂

∂xk

( ∗∂

∂xi

)

=

=∂2

∂xi∂xk+

∂

∂xi

(vk

c2 − w∂

∂t

)

+vi

c2 − w∂2

∂t ∂xk+

+vi

c2−w∂

∂t

(vk

c2−w∂

∂t

)

−∂2

∂xk∂xi−

∂

∂xk

(vi

c2−w∂

∂t

)

−

−vk

c2−w∂2

∂t ∂xi−

vkc2−w

∂

∂t

(vi

c2−w∂

∂t

)

=1

c2−w∂vk∂xi

∂

∂t+

+vk

(c2 − w)2∂w

∂xi∂

∂t+

vkc2 − w

∂2

∂xi∂t+

vic2 − w

∂2

∂t ∂xk+

+vi

(c2 − w)2∂vk∂t

∂

∂t+

vivk(c2 − w)3

∂w

∂t

∂

∂t+

vivk(c2 − w)2

∂2

∂t2−

−1

c2 − w∂vi∂xk

∂

∂t−

vi(c2 − w)2

∂w

∂xk∂

∂t−

vic2 − w

∂2

∂xk∂t−

−vk

c2 − w∂2

∂t ∂xi−

vk(c2 − w)2

∂vi∂t

∂

∂t−

vkvi(c2 − w)3

∂w

∂t

∂

∂t−

2.6 Changing the order of chr.inv.-differentiation 73

−vkvi

(c2 − w)2∂2

∂t2=

[1

c2 − w

(∂vk∂xi

−∂vi∂xk

)

+vk

(c2 − w)2×

×

(∂w

∂xi−∂vi∂t

)

−vi

(c2 − w)2

(∂w

∂xk−∂vk∂t

)]∂

∂t=

=

[(∂vk∂xi

−∂vi∂xk

)

+vk

c2 − w

(∂w

∂xi−∂vi∂t

)

−vi

c2 − w×

×

(∂w

∂xk−∂vk∂t

)]1

c2

∗∂

∂t.

(6.4)

Introducing the notations

Fi =c2

c2 − w

(∂w

∂xi−∂vi∂t

)

, (6.5)

Aik =1

2

(∂vk∂xi

−∂vi∂xk

)

+1

2c2(Fi vk − Fk vi) , (6.6)

we can write∗∂2

∂xi∂t−

∗∂2

∂t ∂xi=1

c2Fi

∗∂

∂t, (6.7)

∗∂2

∂xi∂xk−

∗∂2

∂xk∂xi=2

c2Aik

∗∂

∂t. (6.8)

Inspection of (6.5) and (6.6), reveals that the quantity Fi is acovariant vector and the quantity Aik is an antisymmetric covarianttensor of the 2nd rank. It is not difficult to verify that the quantitiesare a chr.inv.-vector and a chr.inv.-tensor, respectively. Let us as-sume that Q is an arbitrary chr.inv.-invariant. So formulae (6.7)and (6.8) are true for this quantity, i. e.

∗∂2Q

∂xi∂t−

∗∂2Q

∂t ∂xi=1

c2Fi

∗∂Q

∂t, (6.9)

∗∂2Q

∂xi∂xk−

∗∂2Q

∂xk∂xi=2

c2Aik

∗∂Q

∂t. (6.10)

The left-hand sides of the equalities consist of chr.inv.-quantities,while the right-hand sides are the products of the chr.inv.-quantity∗∂Q∂t

and the quantities Fi and Aik, respectively. Hence, Fi and Aikare chr.inv.-quantities as well. Because they contain the potentialsw and vi, we will refer to them as the “power quantities” — thepower vector Fi and the power tensor Aik.


§2.7 The power quantities

Let us suppose thatw′ = 0 (7.1)

is true everywhere inside a four-dimensional volume in a coordi-nate system S′. Then, in another coordinate frame S of the samereference frame, we have

g00

(∂x0

′

∂x0

)2= 1 . (7.2)

The inverse is also true: formula (7.1) is a consequence of (7.2).Thus, we can always set the scalar potential to zero throughoutthe given four-dimensional volume, if we introduce a new timecoordinate x0

′in accordance with the condition (7.2).

Setting the power vector to zero in the given four-dimensionalvolume in the given reference frame is the necessary and sufficientcondition for making the scalar potential and the derivative fromthe vector potential with respect to time, zero (throughout thevolume).

Really, if we have

w ≡ 0 ,∂vi∂t

≡ 0 (7.3)

under a choice of time coordinate, then

Fi ≡ 0 . (7.4)

Conversely, let us assume that the condition (7.4) holds. Then,introducing time coordinate x0 in accordance with the first of theequalities (7.3), we obtain the second of them as a consequence ofthe condition (7.4).

Setting the power vector to zero in the given four-dimensionalvolume in the reference frame is the necessary and sufficient con-dition for making the vector potential zero (throughout the volume).Really, we assume that a coordinate frame

xi′= x0

′(x0, x1, x2, x3)

xi′= xi

′

}

, (7.5)

exists, where we havevi ≡ 0 . (7.6)

2.7 The power quantities 75

Then, everywhere inside the volume, we have

g00 = g00

(∂x0

∂x0

)2, g0i = g00

∂x0

∂x0∂x0

∂xi, (7.7)

thus∂x0

∂x0=c2 − wc2 − w

,∂x0

∂xi= −

c vic2 − w

. (7.8)

On the other hand

dx0 =∂x0

∂x0dx0 +

∂x0

∂xidxi . (7.9)

Therefore we can write

dx0 =(c2 − w)dx0 − cvidxi

c2 − w(7.10)

or, introducing

t =x0

c, t =

x0

c, (7.11)

in the alternative form

dt =(c2 − w)dt− cvidxi

c2 − w. (7.12)

In order that dt exists, it is necessary and sufficient to realizethe exact conditions of the total integrability of the Pfaffian equation

− (c2 − w)dt+ vidxi = 0 . (7.13)

In general, Pfaffian equations

Ndu+ Pdx+Qdy +Rdz = 0 (7.14)

can be totally integrable under the necessary and sufficient condi-tions (any three of the four equations below can be accepted as theconditions∗),

N

(∂Q

∂x−∂P

∂y

)

+ P

(∂N

∂y−∂Q

∂u

)

+Q

(∂P

∂u−∂N

∂x

)

≡ 0

N

(∂R

∂y−∂Q

∂z

)

+Q

(∂N

∂z−∂R

∂u

)

+R

(∂Q

∂u−∂N

∂y

)

≡ 0

N

(∂P

∂z−∂R

∂x

)

+R

(∂N

∂x−∂P

∂u

)

+ P

(∂R

∂u−∂N

∂z

)

≡ 0

, (7.15)

∗The fourth equation is a consequence of the other three.


P

(∂R

∂y−∂Q

∂z

)

+Q

(∂P

∂z−∂R

∂x

)

+R

(∂Q

∂x−∂P

∂y

)

≡ 0 . (7.16)

Assuming

N = −(c2 − w) , P = v1 , Q = v2 , R = v3 , (7.17)

u = t , x = x1, y = x2, z = x3 (7.18)

in the equations, after dividing them by 2(c2−w), we obtain

Aik ≡ 0 . (7.19)

Note that the condition (7.16) take the form

A12v3 + A23v1 + A31 v2 ≡ 0 . (7.20)

It is evident that the simultaneous conditions Fi≡ 0 (7.4) andAik≡ 0 (7.19) are the necessary and sufficient conditions for makingthe scalar potential and the vector potential both zero through-out the four-dimensional volume. The condition (7.19) provides ameans by which the vector potential vi becomes zero. In addition,because of the condition (7.4), the scalar potential w becomes afunction of only the time coordinate t. Then we introduce a newtime coordinate t, which is

dt =(1−

w

c2

)dt . (7.21)

§2.8 The space metric

In any reference frame we have

ds2 = g00(dx0)2 + 2g0idx

0dxi + gik dxidxk (8.1)

under an arbitrary coordinate of time. Going to a new time coor-dinate x0, we set the potentials to zero at the world-point we areconsidering. As a result we obtain

ds2 = (dx0)2 − dσ2, (8.2)

wheredσ2 = −gik dx

idxk (8.3)

is, evidently, the square of a spatial linear element. Let us trans-form (8.3) to an arbitrary time coordinate x0. At the world-point

2.8 The space metric 77

we are considering, we have

g00

(∂x0

∂x0

)2= 1

g00∂x0

∂x0∂x0

∂xi+ g0i

∂x0

∂x0= 0

g00∂x0

∂xi∂x0

∂xk+ g0i

∂x0

∂xk+ g0k

∂x0

∂xi+ gik = gik

. (8.4)

Eliminating ∂x0

∂x0, ∂x

0

∂xi, ∂x

0

∂xkfrom (8.4), we obtain

gik = gik −g0ig0kg00

. (8.5)

Hence, generally speaking∗,

dσ2 =

(

−gik +g0ig0kg00

)

dxidxk. (8.6)

Thus we can introduce the covariant metric sub-tensor hik

dσ2 = hik dxidxk, (8.7)

hik = −gik +g0ig0kg00

, (8.8)

where we havegik = −hik +

vivkc2

. (8.9)

Formula (8.5) shows that hik must be invariant with respect tothe transformations (1.14). We can also see this fact directly, thus


=

= −

(

g′00∂x0

′

∂x0∂x0

′

∂x0+ g′0i

∂x0′

∂xk+ g′0k

∂x0′

∂xi+ g′ik

)

+

+

(

g′00∂x0

′

∂x0∂x0

′

∂x0+ g′0i

∂x0′

∂x0

)(

g′00∂x0

′

∂x0∂x0

′

∂x0+ g′0k

∂x0′

∂x0

)

g′00

(∂x0

′

∂x0

)2 =

∗This equation can also be obtained from other considerations — for instance,see [64], p. 200–201.


= −g′00∂x0

′

∂xi∂x0

′

∂xk− g′0i

∂x0′

∂xk− g′0k

∂x0′

∂xi− g′ik +

+1

g′00

(∂x0

′

∂x0

)2

[(

g′00∂x0

′

∂x0

)2∂x0

′

∂xi∂x0

′

∂xk+

+ g′00g′0i

(∂x0

′

∂x0

)2∂x0

′

∂xk+ g′00g

′0k

(∂x0

′

∂x0

)2∂x0

′

∂xi+

+ g′0ig′0k

(∂x0

′

∂x0

)2∂x0

′

∂xi∂x0

′

∂xk

]

= − g′ik +g′0ig

′0k

g′00= h′ik .

(8.10)

Hence, the covariant metric sub-tensor hik is the chr.inv.-metrictensor. It is evident that the contravariant metric sub-tensor hik,the components of which are given by adjuncts of the determinant

h =

∣∣∣∣∣∣∣

h11 h12 h13

h21 h22 h23

h31 h32 h33

∣∣∣∣∣∣∣, (8.11)

divided by h, is also a chr.inv.-tensor. Because the determinants(8.11) and

g =

∣∣∣∣∣∣∣∣∣

1 0 0 0

0 g11 g12 g13

0 g21 g22 g23

0 g31 g32 g33

∣∣∣∣∣∣∣∣∣

(8.12)

are different only in their signs, then their adjuncts, derived fromthe matrix elements with the same indices i, k=1, 2, 3, are equal.So we have

hik = −gik. (8.13)

Formula (8.13) contains chr.inv.-tensors in both parts. Hence, ingeneral, we have

hik = −gik. (8.14)

We can also introduce the mixed metric sub-tensor hki , which isalso a chr.inv.-tensor,

hki = +gki . (8.15)

2.8 The space metric 79

Let us find relations of h to world-quantities. It is known that∣∣∣∣∣∣∣∣∣∣

g00 g01 g02 g03

g10 g11 g12 g13

g20 g21 g22 g23

g30 g31 g32 g33

∣∣∣∣∣∣∣∣∣∣

=1

g, (8.16)

∣∣∣∣∣∣∣

h11 h12 h13

h21 h22 h23

h31 h32 h33

∣∣∣∣∣∣∣=1

h, (8.17)

and also that the quantities gμν and hik equal to the adjuncts ofthe determinants of gμν (3.16) and hik (3.17), multiplied by g or h,respectively. Then, because of (8.14), we have

g00 = −g

h, (8.18)

or, in the alternative form,√−g =

(1−

w

c2

)√h , (8.19)

where h, as is evident, is not sub-invariant, but a chr.inv.-quantity.Using the chr.inv.-metric tensors, we can contract, substitute,

raise and lower significant indices invariant with respect to the tra-nsformations (1.14). In this way we can transform chr.inv.-quantitiesinto chr.inv.-quantities, and co-quantities into co-quantities∗. Thus,we can build the contravariant vector-potential (the co-vector)

vi = hijvj (8.20)

and the square of the length of the vector-potential itself (the co-invariant)

vivi = hik v

ivk = hikvivk . (8.21)

Because ofg00g

0i + g0jgji = gi0 = 0 , (8.22)

g00g00 + g0jg

j0 = g00 = 1 , (8.23)

we have

g0i = −1

1− wc2

vi

c, (8.24)

∗An exception is the case where a co-quantity vanishes after its contraction.


g00 =1

(1− w

c2

)2

(

1−vj v

j

c2

)

. (8.25)

It is also possible to introduce operations of general covariantdifferentiation. The operations are non-invariant with respect tothe transformations of the time coordinate. For this reason we willintroduce chr.inv.-differentiation, which, being the generalizationof general covariant differentiation, is invariant with respect tothe aforementioned transformations of time. In this process wewill need to differentiate hik, hik, and h with respect to the timecoordinate. Therefore, we will first clarify the kinematic sense ofthose quantities, which are derived from this differentiation. Thefirst such quantity is chr.inv.-velocity, which will be introduced inthe next section.

§2.9 The chr.inv.-vector of velocity

We assume that a point-mass moves with respect to a given refer-ence frame. The vector of its velocity with respect to this referenceframe is

ui =dxi

dt= c

dxi

dx0(9.1)

which, as it is easy to see, is a co-vector. Let us take a world-pointon the world-line of this particle. Applying the necessary choice oftime coordinate x0, we set the potentials to zero at the point. Next,we transform the velocity co-vector

ui = cdxi

dx0, (9.2)

calculated in the time coordinate x0, into an arbitrary time coordi-nate x0. Because

dx0 =∂x0

∂x0dx0 +

∂x0

∂xjdxj , (9.3)

g00 =

(∂x0

∂x0

)2, g0i =

∂x0

∂x0∂x0

∂xi, (9.4)

we havedx0 =

√g00 dx

0 +g0j√g00

dxj , (9.5)

dxi

dx0=

1√g00 +

g0j√g00

dxj

dx0

dxi

dx0, (9.6)

2.9 The chr.inv.-vector of velocity 81

ui =c2ui

c2 − w − vjuj. (9.7)

Equation (9.6) has a chr.inv.-quantity on its left side, so thisquantity, and also equation (9.7), must be invariant with respect tothe transformations (1.14). Really, because

∂x0

∂x0=

∂x0

∂xα′∂xα′

∂x0=∂x0

∂x0′∂x0

′

∂x0= 1

∂x0

∂xj=

∂x0

∂xα′∂xα′

∂xj=∂x0

∂x0′∂x0

′

∂xj+∂x0

∂xj′ = 0

, (9.8)

g′00 = g00

(∂x0

∂x0′

)2, g′0j = g00

∂x0

∂x0′∂x0

∂xj′ + g0j

∂x0

∂x0′, (9.9)

we obtain

1√g′00 +

g′0j√g′00

dxj′

dx0

dxi′

dx0′=

√g′00 dx

i′

g′00dx0′ + g′0jdx

j ′=

=

√

g00

(∂x0

∂x0′

)2dxi

g00

(∂x0

∂x0′

)2(∂x0

′

∂x0dx0+∂x

0′

∂xjdxj)

+

(

g00∂x0

∂x0′∂x0

∂xj′+g0j

∂x0

∂x0′

)

dxj

=

=

√g00 dx

i

g00∂x0

∂x0′∂x0

′

∂x0dx0 + g00

(∂x0

∂x0′∂x0

′

∂xj+ ∂x0

∂xj′

)

dxj + g0jdxj

=

=

√g00 dx

i

g00dx0 + g0jdxj=

1√g00 +

g0j√g00

dxj

dx0

dxi

dx0. (9.10)

Hence, we can introduce the chr.inv.-velocity vector, in brief —the chr.inv.-velocity

∗ui ≡c2ui

c2 − w − vjuj. (9.11)

The chr.inv.-velocity can also be introduced from other consi-derations. Let us introduce the covariant differential of time∗

dx0 = g0αdxα = g00dx

0 + g0jdxj . (9.12)

∗This quantity, generally speaking, is not a total differential.


Because dx0√g00

is a chr.inv.-invariant (see §2.3), the quantity

√g00

dxi

dx0=

√g00 dx

i

g00dx0 + g0jdxj(9.13)

is a chr.inv.-vector: the chr.inv.-velocity we have introduced, div-ided by c.

§2.10 The chr.inv.-tensor of the rate of space deformations

Considering deformations of a continuous medium in any givenreference frame, we introduce the covariant three-dimensional ten-sor of the rate of its deformations Δik in the regular way, i. e.

2Δik = ∇i uk +∇k ui (10.1)

or, in the component form,

2Δik = hkl∂ul

∂xi+ hil

∂ul

∂xk+ (Δil,k +Δkl,i)u

l, (10.2)

where Δpl,q are three-dimensional Christoffel symbols (the sub-symbols) of the 1st kind

Δpl,q =1

2

(∂hpq∂xl

+∂hlq∂xp

−∂hpl∂xq

)

. (10.3)

This sub-tensor of the deformation rates, as is easy to see, isa co-tensor. However, we can introduce the chr.inv.-tensor of thedeformation rates ∗Δik, replacing the co-velocity with the chr.inv.-velocity, and also co-differentiation with chr.inv.-differentiation,respectively. So we have

2∗Δik = hkl∗∂ ∗ul

∂xi+ hil

∗∂ ∗ul

∂xk+ (∗Δil,k +

∗Δkl,i)∗ul, (10.4)

where we denote∗

∗Δpl,q =1

2

( ∗∂hpq∂xl

+∗∂hlq∂xp

−∗∂hpl∂xq

)

. (10.5)

∗Zelmanov subsequently mentioned that the Christoffel sub-symbols (10.3) areaan unnecessary intermediate stage of the algebra. For this reason his scientificarticles of the 1960’s contain non-asterisk notations for the chr.inv.-Christoffelsymbols. For instance, Zelmanov denotes the chr.inv.-Christoffel symbols of the 1stkind (10.5) simply by Δpl,q . We retain the old notation here, for obvious reasons. —Editor’s comment. D. R.

2.11 Deformations of a space 83

We will have particular interest in the case where

∗ul ≡ 0 (10.6)

at the point we are considering. Let us introduce the followingspecial notation for the chr.inv.-tensor of the deformation rates

∗Δik = Dik . (10.7)

In this case∗ we haveul ≡ 0 , (10.8)

∗∂ ∗ul

∂xs=

c2

c2 − w∂ul

∂xs, (10.9)

2Dik =c2

c2 − w

(

hkl∂ul

∂xi+ hil

∂ul

∂xk

)

. (10.10)

Hence, it is clear that the equality (10.10) holds in all coordinateframes of this reference frame.

Introducing the coordinates

x0 = x0 (x0, x1, x2, x3)

xi = xi

}

, (10.11)

wherew = 0 , (10.12)

we will have,

2Dik = hkl∂ul

∂xi+ hil

∂ul

∂xk. (10.13)

On the other hand, we can write the co-tensor of the deforma-tion rates under the condition ul≡ 0 (10.8), in general, as follows

2Δik = hkl∂ul

∂xi+ hil

∂ul

∂xk, (10.14)

and so we haveDik = Δik . (10.15)

§2.11 Deformations of a space

Let us consider a volume around an arbitrary point a in the spaceof coordinates

xi = ai , ai = consti. (11.1)

∗Zelmanov assumes that the derivative of ui can be finite and essentially non-zero under ui→ 0. He also assumes that this quantity is static, i. e. ui=0. — Editor’scomment. D. R.


Taking a small time interval, we can make this volume so smallthat any of its points at any moment of time (inside the timeinterval) has the univalent defined geodesic distance σ to the pointa. The quantity σ2 differs from (hpq)a (x

p− ap)(xq − aq) with highorder terms in xi− ai sufficiently small quantities. So, with thevolume around the point a sufficiently small, we can write

σ2 =[(hpq)a + αpq,j(x

j − aj)](xp− ap)(xq − aq) , (11.2)

where αpq,j are finite∗ (as it easy to see, we can assume αpq,jsymmetric with respect to its indices p and q). Generally speaking,hik are functions of t (the space undergoes deformations), so thegeodesic distances between the given points in the space and thepoint a change with time. Let us also introduce an auxiliary refer-ence frame, defining it by the conditions: (1) this auxiliary systemis fixed with respect to the initial reference frame of the point a insuch a way that if ∗ui is the chr.inv.-velocity of the auxiliary systemwith respect to the initial system (the ∗ui is measured in the initialsystem), then at the point a we have

∗ui ≡ 0 ; (11.3)

and (2) the geodesic distances (measured in the initial system)between the point a and all given points of the auxiliary systemnear a, remain unchanged, so

∗∂σ2

∂t= 0 . (11.4)

We will refer to this auxiliary system as the locally-stationaryreference frame at the point a.

It is evident that the locally-stationary at the point a system isnot uniquely defined, but in the order of its arbitrary rotation nearthe point. The following speculations are related to any referenceframe of infinitely numerous system,which are locally-orthogonallyat this point.

Formula (11.4), because of (11.2), gives[∗∂(hpq)a

∂t+∗∂αpq,j∂t

(xj−aj)+c2αpq,jc2−w

uj]

(xp−ap)(xq−aq)+

+ 2c2(hpq)a+αpq,j (x

j−aj)c2−w

up(xq−aq) = 0 ,

(11.5)

∗We suppose that the derivatives we are considering exist and are finite.


whereui =

dxi

dt(11.6)

is the velocity of the space (measured at the point xi in the refer-ence frame), which is locally-stationary at the point a with respectto the space. Introducing

Θpq =∗∂(hpq)a

∂t+

∗∂αpq,j∂t

(xj − aj) +c2αpq,jc2 − w

uj , (11.7)

Ξpq = 2c2 (hpq)a + αpq,j(x

j − aj)c2 − w

, (11.8)

we can re-write (11.5) in the form

Θpq(xp − ap)(xq − aq) + Ξpqu

p(xq − aq) = 0 . (11.9)

Differentiating this equality term-by-term twice (with respectto xk and with respect to xi, respectively), we obtain

∂2Θpq∂xi∂xk

(xp−ap)(xq−aq)+2

(∂Θkq∂xi

+∂Θiq∂xk

)

(xq−aq)+

+2Θik+∂2Ξpq∂xi∂xk

up(xq−aq)+

(∂Ξpq∂xk

∂up

∂xi+∂Ξpq∂xi

∂up

∂xk

)

×

× (xq−aq)+Ξpq∂2up

∂xi∂xk(xq−aq)+

(∂Ξiq∂xk

+∂Ξkq∂xi

)

uq +

+

(

Ξiq∂uq

∂xk+Ξkq

∂uq

∂xi

)

=0 .

(11.10)

At the point a, taking the first of the equalities (11.3) intoaccount, we have

ui = 0 , (11.11)then

2 (Θik)a + (Ξiq)a

(∂uq

∂xk

)

a

+ (Ξkq)a

(∂uq

∂xi

)

a

= 0 . (11.12)

In other words, because

(Θik)a =∗∂ (hik)a

∂t, (11.13)

(Ξik)a = 2c2 (hik)ac2 − (w)a

, (11.14)


we have∗∂ (hik)a

∂t+

c2

c2−(w)a

[

(hkq)a

(∂uq

∂xi

)

a

+(hiq)a

(∂uq

∂xk

)

a

]

= 0 (11.15)

or, in the final form∗∂ (hik)a

∂t= −

[c2

c2 − w

(

hkq∂uq

∂xi+ hiq

∂uq

∂xk

)]

a

. (11.16)

We now introduce the chr.inv.-tensor of the deformation rate ofthe space of the reference frame (defined in this reference frame)with respect to the space, which is locally-stationary at the pointa. We assume that

2∗Δik = hkq∗∂ ∗uq

∂xi+ hiq

∗∂ ∗uq

∂xk+ (∗Δiq,k +

∗Δkq,i)∗uq, (11.17)

where ∗uq is the chr.inv.-velocity of the space (measured in thereference frame) with respect to the locally stationary space at thepoint a. It is evident that

∗uq = −∗uq, (11.18)

where ∗uq is the chr.inv.-velocity of the locally-stationary space atthe point a, measured with respect to the space. Hence

2∗Δik = −

(

hkq∗∂ ∗uq

∂xi+ hiq

∗∂ ∗uq

∂xk

)

− (∗Δiq,k +∗Δkq,i)

∗uq (11.19)

characterizes deformations of the space at any point near a. For-mula (11.3) holds at the point a, so as well as (10.10) we obtain

2 (∗Δik)a = 2Dik = −

[c2

c2 − w

(

hkq∂uq

∂xi+ hiq

∂uq

∂xk

)]

a

. (11.20)

We have thus obtained the chr.inv.-tensor of the deformation ra-tes of the space at the point a with respect to the locally-stationaryspace at the point a. Comparing (11.20) and (11.16), we obtain

∗∂ (hik)a∂t

= 2 (Dik)a . (11.21)

Because the point a is arbitrary, we can, in general, write∗

∗∂hik∂t

= 2Dik . (11.22)

∗Zelmanov called equation (11.22) the theorem on the space deformations,see §2.13. — Editor’s comment. D. R.


Thus, at any point of the space, the chr.inv.-derivative of the co-variant chr.inv.-metric tensor with respect to time is twice the co-variant chr.inv.-tensor of the deformation rates of this space (withrespect to the locally-stationary space at this point).

Let us consider∗∂hik

∂t. Because

hijhjk = hki = δki , (11.23)

we have

hij∗∂hjk

∂t= −

∗∂hij∂t

hjk, (11.24)

and finally, taking (11.22) into account, we obtain

hij∗∂hjk

∂t= −2Dk

i . (11.25)

Raising the index i, we obtain

∗∂hik

∂t= −2Dik. (11.26)

Thus, in any point of the space, the chr.inv.-derivative of thecontravariant chr.inv.-metric tensor with respect to time is twicethe negative of the contravariant chr.inv.-tensor of the deformationrates of this space (with respect to the locally-stationary space atthis point).

Let us consider∗∂h∂t

. Applying the rule for differentiating det-

erminants and taking (11.22) into account, gives

∗∂h

∂t=

∣∣∣∣∣∣∣

2D11 h12 h13

2D21 h22 h23

2D31 h32 h33

∣∣∣∣∣∣∣+

∣∣∣∣∣∣∣

h11 2D12 h13

h21 2D22 h23

h31 2D32 h33

∣∣∣∣∣∣∣+

∣∣∣∣∣∣∣

h11 h12 2D13

h21 h22 2D23

h31 h32 2D33

∣∣∣∣∣∣∣=

= 2h(Di1h

i1 +Di2hi2 +Di3h

i3)= 2hhikDik . (11.27)

Introducing the chr.inv.-invariant of the deformation rates atthe point (with respect to the locally-stationary space at this point)

D = hikDik = hikDik = D

jj , (11.28)

we obtain1

h

∗∂h

∂t= 2D . (11.29)


Thus, at any point of the space, the derivative of the logarithmof the fundamental determinant with respect to time is twice thechr.inv.-invariant of the deformation rates of this space (with re-spect to the locally-stationary space at this point).

Clearly, the fact that the locally-stationary reference system atthe point is not unique does not affect our conclusions.

Furthermore, dropping the terms “locally-stationary space” and“its reference systems” for brevity, we will merely refer to thespace deformations.

§2.12 Transformations of space elements

We will now deduce numerous equations describing deformationsof space by analogy with equations for deformations of continuousmedia.

We assume that δL1a is the length of an elementary coordinateinterval on the x1 axis, i. e.

δL1a =√h11

∣∣δax

1∣∣ , δax

1 = const1a , (12.1)

then (for instance, see [62], p. 365)

∗∂(δL1a)∂t

=1

2√h11

∗∂h11∂t

∣∣δax

1∣∣ =

D11√h11

∣∣δax

1∣∣ , (12.2)

1

δL1a

∗∂(δL1a)∂t

=D11h11

. (12.3)

Thus, D11

h11is the rate of the relative lengthening (because of the

space deformations) of the linear element along the x1 axis.Next, we assume that δS23ab is the square of an element of the

coordinate surface x2, x3

δS23ab =

√√√√∣∣∣∣∣

h22 h23

h32 h33

∣∣∣∣∣

∣∣δΠ23ab

∣∣

δΠ23ab =

∣∣∣∣∣

δax2 δax

3

δbx1 δbx

2

∣∣∣∣∣, δax

i = constia , δbxi = constib

, (12.4)

2.13 Christoffel’s symbols in chr.inv.-form 89

then we have

∗∂(δS23ab

)

∂t=

1

2√hh11

( ∗∂h

∂th11 + h

∗∂h11

∂t

) ∣∣δΠ23ab

∣∣ =

=Dhh11 − hD11

√hh11

∣∣δΠ23ab

∣∣ ,

(12.5)

1

δS23ab

∗∂(δS23ab

)

∂t= D −

D11

h11. (12.6)

Thus, D− D11

h11is the rate of the relative expansion (because of

the space deformations) of the element of the surface x2, x3.Finally, we assume that δVabc is the value of a volume element

δVabc =√h∣∣δΠ123abc

∣∣

δΠ123abc =

∣∣∣∣∣∣∣

δax1 δax

2 δax3

δbx1 δbx

2 δbx3

δcx1 δcx

2 δcx3

∣∣∣∣∣∣∣

δaxi = constia

δbxi = constib

δcxi = constic

. (12.7)

Because

∗∂ (δVabc)

∂t=

1

2√h

∗∂h

∂t

∣∣δΠ123abc

∣∣ =

hD√h

∣∣δΠ123abc

∣∣ , (12.8)

we have (for instance, see [62], p. 366)

1

δVabc

∗∂ (δVabc)

∂t= D . (12.9)

So the chr.inv.-invariant D is the rate of the relative expansionsof the volume element (because of the space deformations).

§2.13 Christoffel’s symbols in chr.inv.-form

We will use two species of Christoffel symbols of the 1st kind, see(10.3) and (10.5); the co-symbols

Δij,k =1

2

(∂hik∂xj

+∂hjk∂xi

−∂hij∂xk

)

(13.1)


and the chr.inv.-symbols

∗Δij,k =1

2

( ∗∂hik∂xj

+∗∂hjk∂xi

−∗∂hij∂xk

)

. (13.2)

In accordance with the foregoing, we will use two species ofChristoffel sub-symbols of the 2nd kind, namely, the co-symbols

Δkij = hklΔij,l (13.3)

and the chr.inv.-symbols∗Δkij = hkl ∗Δij,l . (13.4)

Let us find formulae for the Christoffel chr.inv.-symbols, em-ploying the theorem on the space deformations (11.22). Because

∗∂

∂xj=

∂

∂xj+vjc2

∗∂

∂t, (13.5)

we have∗∂hik∂xj

=∂hik∂xj

+2

c2Dik vj . (13.6)

Therefore, we obtain

∗Δij,k = Δij,k +1

c2(Dikvj +Djkvi −Dijvk) , (13.7)

∗Δkij = Δkij +

1

c2(Dki vj +D

kj vi −Dijv

k). (13.8)

It is necessary to highlight various properties of the chr.inv.-Christoffel symbols corresponding to the analogous properties ofthe co-symbols. The first of these is their symmetry

∗Δij,k =∗Δji,k , (13.9)

∗Δkij =∗Δkji . (13.10)

Then, considering (13.2), we have

∗Δij,k +∗Δkj,i =

∗∂hik∂xj

. (13.11)

Finally, because

Δjij +

1

c2

(Dji vj +D

jj vi −Dijv

j)=∂ ln

√h

∂xi+vic2D , (13.12)

and taking (13.8) and (13.5) into account, we obtain

∗Δjij =

∗∂ ln√h

∂xi. (13.13)

2.14 General covariant differentiation in chr.inv.-form 91

§2.14 General covariant differentiation in chr.inv.-form

We now introduce operations of chr.inv.-differentiation, definingthe operations by the requirements∗: (1) they must be invariantwith respect to the transformations of the time coordinate; and(2) they must coincide with the operations of regular covariantdifferentiation when the potentials are set to zero. To realize therequirements in the operations of regular covariant differentiation,it is necessary and sufficient to replace all co-derivatives withchr.inv.-derivatives and, respectively, to replace the usual symbolsChristoffel with the chr.inv.-Christoffel symbols.

Modifying the symbol ∇ denoting regular covariant different-iation, we denote chr.inv.-differentiation by ∗∇. Accordingly, wehave the chr.inv.-differential operations for sub-vectors

∗∇iQk =∗∂Qkdxi

− ∗ΔlikQl , (14.1)

∗∇iQk =

∗∂Qk

dxi+ ∗ΔkilQ

l, (14.2)

for sub-tensors of the 2nd rank

∗∇iQjk =∗∂Qjkdxi

− ∗ΔlijQlk −∗ΔlikQjl , (14.3)

∗∇iQkj =

∗∂Qkjdxi

− ∗ΔlijQkl +

∗ΔkilQlj , (14.4)

∗∇iQjk =

∗∂Qjk

dxi+ ∗Δ

jilQ

lk + ∗ΔkilQjl, (14.5)

and so forth. In general, chr.inv.-derivative notation is different toregular covariant derivative notation merely by an asterisk beforetheir ∇ and Christoffel symbols.

It is clear that the chr.inv.-derivative of a sub-tensor quantitywill be a chr.inv.-quantity if the differentiated sub-tensor is also achr.inv.-quantity.

∗Zelmanov uses two different terms here, denoting chronometrically invariantdifferential operations, namely — chr.inv.-derivation for chr.inv.-derivatives withrespect to time and spatial coordinates (see §2.5), and chr.inv.-covariantdifferentiation, generalizing covariant four-dimensional differentiation into itsthree-dimensional analogue, which has the property of chronometric invariance.Later, Zelmanov did not use the separate terms, because it was unnecessary, and soonly used the term “chr.inv.-differentiation” for any differential operations whichhave the property of chronometric invariance. — Editor’s comment. D. R.


Regular divergence can be defined as a covariant derivative con-tracted with one of the upper indices of the differentiated quantity.For this reason we will refer to the quantity, obtained from chr.inv.-differentiation in this way, as chr.inv.-divergence. For instance,

∗∇iQi =

∗∂Qi

∂xi+

∗∂ ln√h

∂xiQi, (14.6)

∗∇iQi =

1√h

∗∂(√hQi

)

∂xi, (14.7)

∗∇iQij =

∗∂Qij∂xi

− ∗ΔlijQil +

∗∂ ln√h

∂xiQij , (14.8)

∗∇iQji =

∗∂Qji

∂xi+ ∗Δ

jilQ

il +∗∂ ln

√h

∂xiQji. (14.9)

In relation to chr.inv.-differentiation, the metric sub-tensors arethe same as for regular covariant differentiation. Really,

∗∇j hik =∗∂hik∂xj

− ∗Δljihlk −∗Δljkhil =

∗∂hik∂xj

− ∗Δji,k−∗Δjk,i (14.10)

and, because of (13.11), we have

∗∇j hik = 0 . (14.11)

We therefore have

∗∇j hki =

∗∂hki∂xj

− ∗Δljihkl +

∗Δkjlhli = −

∗Δkji +∗Δkji , (14.12)

and thus obtain∗∇j h

ki = 0 . (14.13)

Becausehki = hqih

qk, (14.14)


hqi∗∇j h

qk = 0 (14.15)

or, raising the index i,

hiq∗∇j h

qk = 0 . (14.16)

Finally, becausehiqh

qk = hik, (14.17)

2.15 The Riemann-Christoffel tensor in chr.inv.-form 93

formula (14.16), taking (14.13) into account, gives

∗∇j hik = 0 . (14.18)

So the operation of chr.inv.-differentiation, as well as the oper-ation of regular covariant differentiation, is commutative with re-spect to the raising, lowering, or substitution of indices.

We can accordingly re-write formula (10.4) as follows

2∗Δik = hkl∗∇i

∗ul + hil∗∇k

∗ul (14.19)

or, in the alternative form

2∗Δik =∗∇i

∗uk +∗∇k

∗ui . (14.20)

These are equations for general covariant differentiation inchr.inv.-form, obtained from the chr.inv.-tensor of the deformationrates as an example. It easy to see that the equations are differentfrom the analogous equations for the co-tensor of the deformationrates only by the presence of an asterisk.

§2.15 The Riemann-Christoffel tensor in chr.inv.-form

Let us denote∗∇pq =

∗∇p (∗∇q ) . (15.1)

We are going to take any sub-vectors Qk and Qk in order tochange the sequence of their chr.inv.-covariant differentiation. Asa result, for Qk, changing its dummy indices, we obtain

∗∇ijQk −∗∇jiQk =

∗∇i (∗∇jQk)−

∗∇j (∗∇iQk) =

=∗∂

∂xi(∗∇jQk)−

∗Δlij (∗∇lQk)−

∗Δlik (∗∇jQl)−

−∗∂

∂xj(∗∇iQk) +

∗Δlji (∗∇lQk) +

∗Δljk(∗∇iQl) =

=∗∂

∂xi

(∗∂Qk∂xj

−∗ΔljkQl

)

−∗∂

∂xj

(∗∂Qk∂xi

−∗ΔlikQl

)

−

− ∗Δlik

( ∗∂Ql∂xj

− ∗ΔmjlQm

)

+ ∗Δljk

( ∗∂Ql∂xi

− ∗ΔmilQm

)

=

=

( ∗∂2Qk∂xi∂xj

−∗∂2Qk∂xj∂xi

)

−

( ∗∂ ∗Δljk∂xi

−∗∂ ∗Δlik∂xj

)

Ql −


−

(∗Δljk

∗∂Ql∂xi

− ∗Δlik

∗∂Ql∂xj

)

−

(∗Δlik

∗∂Ql∂xj

− ∗Δljk

∗∂Ql∂xi

)

+

+(∗Δlik

∗Δmjl −∗Δljk

∗Δmil)Qm =

2

c2Aij

∗∂Qk∂t

+

+

( ∗∂ ∗Δlik∂xj

−∗∂ ∗Δljk∂xi

+ ∗Δmik∗Δljm −

∗Δmjk∗Δlim

)

Ql .

(15.2)

Doing the same for the contravariant sub-vector Qk, we have

∗∇ijQk − ∗∇jiQ

k = ∗∇i(∗∇jQ

k)− ∗∇j

(∗∇iQ

k)=

=∗∂

∂xi(∗∇jQ

k)− ∗Δlij

(∗∇lQ

k)+ ∗Δkil

(∗∇jQ

l)−

−∗∂

∂xj(∗∇iQ

l)+ ∗Δlji

(∗∇lQ

k)− ∗Δkjl

(∗∇iQ

l)=

=∗∂

∂xi

(∗∂Qk

∂xj+ ∗ΔkjlQ

l

)

−∗∂

∂xj

(∗∂Qk

∂xi+ ∗ΔkilQ

l

)

+

+ ∗Δkil

( ∗∂Ql

∂xj+ ∗ΔljmQ

m

)

−∗Δkjl

( ∗∂Ql

∂xi+ ∗ΔlimQ

m

)

=

=

( ∗∂2Qk

∂xi∂xj−

∗∂2Qk

∂xj∂xi

)

−

( ∗∂ ∗Δkjl∂xi

−∗∂ ∗Δkil∂xj

)

Ql +

+

(∗Δkjl

∗∂Ql

∂xi− ∗Δkil

∗∂Ql

∂xj

)

+

(∗Δkil

∗∂Ql

∂xj− ∗Δkjl

∗∂Ql

∂xi

)

+

+(∗Δkil

∗Δljm −∗Δkjl

∗Δlim)Qm =

2

c2Aij

∗∂Qk

∂t−

−

( ∗∂ ∗Δkil∂xj

−∗∂ ∗Δkjl∂xi

+ ∗Δmil∗Δkjm −

∗Δmjl∗Δkim

)

Ql.

(15.3)

Introducing the notation

H ∙∙∙lkji =

∗∂ ∗Δlik∂xj


+ ∗Δmik∗Δljm −

∗Δmjk∗Δlim , (15.4)

we can write

∗∇ijQk −∗∇jiQk =

2

c2Aij

∗∂Qk∂t

+H ∙∙∙lkjiQl , (15.5)

∗∇ijQk − ∗∇jiQ

k =2

c2Aij

∗∂Qk

∂t−H ∙∙∙k

lji Ql. (15.6)

2.15 The Riemann-Christoffel tensor in chr.inv.-form 95

Because Qk (and Qk, respectively) is an arbitrary sub-vector,then, according to the theorem of fractions, H ∙∙∙l

kji is a sub-tensorof the 4th rank, which is thrice covariant and once contravariant.Since Qk (and Qk) is an arbitrary chr.inv.-vector, we are assuredthat H ∙∙∙l

kji is a chr.inv.-tensor. By the relationship of its structureto the structure of the Riemann-Christoffel mixed sub-tensor (theco-tensor)

K ∙∙∙lkji =

∂Δlik∂xj

−∂Δljk∂xi

+ΔmikΔljm −Δ

mjkΔ

lim , (15.7)

as a base, we call H ∙∙∙lkji the chr.inv.-Riemann-Christoffel mixed

tensor. Lowering the upper index and using (13.11)

hnlH∙∙∙lkji = hnl

(∗∂ ∗Δlik∂xj


+∗Δmik∗Δljm−

∗Δmjk∗Δlim

)

=

=∗∂ ∗Δik,n∂xj

− (∗Δnj,l +∗Δ lj,n)

∗Δlik −∗∂ ∗Δjk,n∂xi

+

+ (∗Δni,l +∗Δli,n)

∗Δljk +∗Δmik

∗Δjm,n −∗Δmjk

∗Δim,n =

=∗∂ ∗Δik,n∂xj

−∗∂ ∗Δjk,n∂xi

− ∗Δnj,l∗Δlik +

∗Δni,l∗Δljk ,

(15.8)

we obtain, after rearrangement of its dummy indices, the chr.inv.-Riemann-Christoffel covariant tensor

Hkjin =∗∂ ∗Δik,n∂xj


− ∗Δik,l∗Δljn +

∗Δjk,l∗Δlin , (15.9)

the structure of which has the relationship to the Riemann-Christ-offel covariant co-tensor∗

Kkjin =∂Δik,n∂xj

−∂Δjk,n∂xi

− Δik,lΔljn + Δjk,lΔ

lin . (15.10)

Some of properties of the chr.inv.-tensor Hkjin are given belowas follows. Because

( ∗∂ ∗Δik,n∂xj


)

+

( ∗∂ ∗Δin,k∂xj

−∗∂ ∗Δjn,k∂xi

)

=

∗Sites of different indices in the Riemann-Christoffel tensor, and also theirmeanings, are different in various publications, because of different notation forthe Riemann-Christoffel tensor (for instance, see [8], p. 91 and [7], p. 130). We arefollowing Eddington here.


=∗∂

∂xj(∗Δik,n +

∗Δin,k)−∗∂

∂xi(∗Δjk,n +

∗Δjn,k) =

=∗∂2hkn∂xj∂xi

−∗∂2hkn∂xi∂xj

=2

c2Aji

∗∂hkn∂t

=4

c2AjiDkn ,

(15.11)

we have1

2(Hkjin +Hnjik) =

2

c2AjiDkn , (15.12)

and also1

2(Hkjin +Hkijn) = 0 . (15.13)

Thus, the chr.inv.-tensor Hkjin as well as Kkjin is antisymmetricwith respect to the inner pair of indices and, in contrast to Kkjin,generally speaking, is antisymmetric with respect to the outer pairof indices. It is possible to show that Hkjin has, generally speaking,no other symmetric properties that are typical for Kkjin. However,to set one of the tensors Aik or Dik to zero will be sufficient forobtaining the tensor Hkjin by the symmetric properties.

We will refer to the chr.inv.-tensor of the 2nd rank

Hkj = H ∙∙∙lkjl = Hkjinh

in, (15.14)

obtained as a result of the contraction of the chr.inv.-Riemann-Christoffel tensor by the second pair of indices, as Einstein’schr.inv.-tensor. Because of (15.4), we have

Hkj =∗∂ ∗Δlkl∂xj

−∗∂ ∗Δlkj∂xl

+ ∗Δmkl∗Δljm −

∗Δmkj∗Δlml , (15.15)


Hkj = −∗∂ ∗Δlkj∂xl

+ ∗Δmkl∗Δljm +

∗∂2 ln√h

∂xj∂xk− ∗Δlkj

∗∂ ln√h

∂xl, (15.16)

which is like the formulae for Einstein’s co-tensor

Kkj = K ∙∙∙lkjl = Kkjinh

in, (15.17)

Kkj =∂Δlkl∂xj

−∂Δlkj∂xl

+ΔmklΔljm −Δ

mkjΔ

lml , (15.18)

Kkj = −∂Δlkj∂xl

+ΔmklΔljm +

∂2 ln√h

∂xj∂xk−Δlkj

∂ ln√h

∂xl, (15.19)

obtained as a result of the contraction of the Riemann-Christoffelco-tensor by the second pair of its indices.

2.16 Chr.inv.-rotor 97

In contrast to Kkj , the chr.inv.-tensor Hkj , generally speaking,is not symmetric. Indeed,

∗∂2 ln√h

∂xj∂xk−

∗∂2 ln√h

∂xk∂xj=2

c2AjkD , (15.20)

so we have1

2(Hkj −Hjk) =

1

c2AjkD . (15.21)

We also introduce the chr.inv.-invariant

H = hkjHkj . (15.22)

Using the chr.inv.-Riemann-Christoffel tensor and also thosequantities result from its contractions, we will consider the problemof space curvature and the problem of space rotations. We beginthese studies from the latter problem, where we will need to intro-duce chr.inv.-angular velocity.

§2.16 Chr.inv.-rotor

Let us introduce by analogy with regular three-dimensional calcu-lus, the contravariant sub-tensor of the 3rd rank εijk, which iscompletely defined by its component

ε123 =1√h

(16.1)

and its antisymmetry with respect to any pair of its indices. We alsointroduce the covariant sub-tensor εijk. The sub-tensor εijk has thesame symmetric properties (for instance, see [8], p. 78), and

ε123 =√h . (16.2)

Because√h is a chr.inv.-quantity, the sub-tensors εijk and εijk

are chr.inv.-tensors as well. These are some of their properties:(1) the tensors are linked the index operators (the components ofthe chr.inv.-metric tensor)

εpqkεijk = hpi h

qj − h

pj h

qi , (16.3)

and (2) the chr.inv.-derivative of them is zero

∗∇p εijk = 0 . (16.4)


Formula (16.3) is known from regular tensor analysis (see [8],p. 111). Formula (16.4) is derived from the fact that the regularcovariant derivative of the tensor εijk is zero (see [8], p. 88)

∇p εijk = 0 . (16.5)

Actually, considering coordinates x0, x1, x2, x3, which set thepotentials to zero at the world-point we are considering, we have

∗∇p εijk = ∇p εijk (16.6)

so, because of (16.5), we obtain

∗∇p εijk = 0 . (16.7)

The left side of the equality (16.7) is chr.inv.-tensor of the 4thrank. Hence, in arbitrary coordinates, it becomes zero — the for-mula (16.4) is true.

Just as in regular tensor algebra, using the chr.inv.-tensors εijk

and εijk, we can set sub-vectors ωk or ψk dual to any antisymmetricsub-tensor of the 2nd rank aij or bij

ωk =1

2εijkaij , (16.8)

ψk =1

2εijk b

ij . (16.9)

The converse is also true. Using the chr.inv.-tensors εijk andεijk, we can uniquely set the antisymmetric tensors of the 2ndrank dual to any sub-vector.

If the sub-tensors aij and bij are mutually conjuncted, thenthe sub-vectors ωk and ψk are mutually conjuncted as well. If aijand bij are chr.inv.-tensors, then ωk and ψk are chr.inv.-vectors.Multiplying (16.8) term-by-term by εpqk, and multiplying (16.9)term-by-term by εpqk, after the results are contracted, we obtain

εpqk ωk =

1

2εijkεpqk aij , (16.10)

εpqkψk =1

2εpqkεijk b

ij , (16.11)

and, because of (16.3), we have

apq = εpqk ωk, (16.12)

2.16 Chr.inv.-rotor 99

bpq = εpqkψk . (16.13)

It is evident that if the sub-vectors ωk and ψk are mutually con-juncted, then the sub-tensors apq and bpq are mutually conjunctedas well. If ωk and ψk are chr.inv.-vectors, then apq and bpq arechr.inv.-tensors. It is clear that formulae (16.8–16.9) and (16.12–16.13) set the unique mutual link between sub-vectors and anti-symmetric dual sub-tensors of the 2nd rank.

The regular definition of the rotor (vortex) rp of a vector ωk is∗

rp (ω) = εqkp∇q ωk . (16.14)

We introduce the chr.inv.-rotor

∗rp (ω) = εqkp ∗∇q ωk . (16.15)

It is evident that the operation of building chr.inv.-rotors is in-variant with respect to transformations of the time coordinate. Thisoperation coincides with the operation of building regular co-rotorswhen the potentials are set to zero at the world-point we are con-sidering. It is also evident that the chr.inv.-rotor of a chr.inv.-vectoris a chr.inv.-vector.

Let us introduce the antisymmetric tensor of the 2nd rank aij

dual to the vector ωk. So we have

ωk =1

2εijk a

ij . (16.16)

Substituting (16.16) into (16.15), we have

∗rp (ω) =1

2εqkpεijk

∗∇q aij +

1

2εqkpaij ∗∇q εijk =

=1

2εqkpεijk

∗∇q aij = −

1

2εqpkεijk

∗∇q aij =

= −1

2

(hqi h

pj − h

qjh

pi

)∗∇q a

ij = −1

2

(∗∇i a

ip−∗∇j apj),

(16.17)

and so∗rp (ω) = ∗∇j a

pj , (16.18)


εqkp ∗∇q ωk =∗∇j a

pj . (16.19)

∗In general, the vortex of a vector is antisymmetric tensor of the 2nd rank. How-ever, in the particular case of three-dimensional space, the vortex can be assumedthe vector dual to the tensor. We also use this representation here (see [8], p. 79–80).


So the chr.inv.-rotor of any sub-vector is the chr.inv.-divergenceof its antisymmetric dual sub-tensor of the 2nd rank.

§2.17 The chr.inv.-vector of the angular velocities of rotation. Itschr.inv-rotor

The regular relation between the contravariant sub-vector ωk of theangular velocities of a volume element and the covariant sub-vectoruj of the linear velocities of its points is

ωk =1

2rk (u) (17.1)


ωk =1

2εijk∇i uj . (17.2)

The sub-vector ωk has the dual antisymmetric sub-tensor

aij =1

2(∇i uj −∇j ui) , (17.3)

and so we have

ωk =1

2εijkaij . (17.4)

It easy to see that the sub-vector (17.2) and the sub-tensor(17.3) are co-vector and co-tensor, respectively. At the same time,we can introduce the chr.inv.-vector of the angular velocities ∗ωk

and its chr.inv.-dual tensor ∗aij , by replacing this co-vector with thechr.inv.-vector and co-differentiation with chr.inv.-differentiationin the equations (17.2) and (17.3)

∗ωk =1

2∗rk (∗u) , (17.5)

∗ωk =1

2εijk ∗∇i

∗uj , (17.6)

∗aij =1

2(∗∇i

∗uj −∗∇j

∗ui) , (17.7)

so we arrive at∗ωk =

1

2εijk ∗aij . (17.8)

Taking the chr.inv.-rotor from the chr.inv.-vector of the angularvelocities, because of (16.18) or (16.19), we obtain

∗ri (∗ω) = ∗∇j∗aij , (17.9)

2.18 Differential rotations. Differential deformations 101


εqki∗∇i∗ωk =

∗∇j∗aqj , (17.10)

where∗aij =

1

2

(hil ∗∇l

∗uj − hjl ∗∇l∗ui). (17.11)

We will have interest in that case where ∗ui≡ 0 (10.6) holdsat the world-point we are considering. Hence, ui≡ 0 (10.8) and∗∂ ∗ui

∂xk= c2

c2−w∂ui

∂xk(10.9) must also hold at the point. In this case

we have∗ωk =

1

2

c2

c2 − wεijkhjl

∂ul

∂xi, (17.12)

ωk =1

2εijkhjl

∂ul

∂xi. (17.13)

Introducing the coordinates (10.11), where the condition w = 0(10.12) is true, this case gives

∗ωk = ωk. (17.14)

In the case we are considering, we introduce the special bondfor the chr.inv.-rotor of the vector of the angular velocities

∗ri (∗ω) = Ri (∗ω) . (17.15)

§2.18 Differential rotations. Differential deformations


∗∇j∗aij =

1

2

(hil ∗∇jl

∗uj − hjl ∗∇jl∗ui). (18.1)

In accordance with (15.6), we can write

∗∇jl∗ui = ∗∇lj

∗ui +2

c2Ajl

∗∂ ∗ui

∂t−H ∙∙∙i

nlj∗un (18.2)

∗∇jl∗uj = ∗∇lj

∗uj +2

c2Ajl

∗∂ ∗uj

∂t−Hnl

∗un. (18.3)

For this reason, we have

∗∇j∗aij =

1

2hil ∗∇lj

∗uj +1

c2A∙ij∙

∗∂ ∗uj

∂t−1

2Hin∗un −


−1

2hjl∗∇lj

∗ui =1

2∗∇l

(hil ∗∇j

∗uj − hjl ∗∇j∗ui)+

+1

c2A∙ij∙

∗∂ ∗uj

∂t−1

2Hin∗un.

(18.4)

At the same time,

∗∇j∗ui = hil ∗∇j

∗ul = hil (∗Δjl +∗ajl) =

∗Δij +∗aij , (18.5)

where ∗Δjl is derived from (10.4). Hence

∗∇j∗uj = ∗Δ , (18.6)

where∗Δ = hik ∗Δik = hik

∗Δik = ∗Δjj . (18.7)

Therefore we arrive at

∗∇j∗aij =

1

2∗∇l(hil∗Δ−∗Δil+∗ail

)+1

c2A∙ij∙

∗∂ ∗uj

∂t−1

2Hin∗un, (18.8)

∗∇j∗aij = ∗∇j

(hij ∗Δ− ∗Δij

)+2

c2A∙ij∙

∗∂ ∗uj

∂t−Hi

j∗uj , (18.9)


∗ri (∗ω) = ∗∇j(hij ∗Δ− ∗Δij

)+2

c2A∙ij∙

∗∂ ∗uj

∂t−Hi

j∗uj . (18.10)

§2.19 Differential rotations of a space

We assume that ∗Δij is the contravariant chr.inv.-tensor of thedeformation rates of the space near the point

xi = ai, (19.1)

measured with respect to the reference frame that is locally-stationary at this point. We can then write

∗Δij = ∗Δij ( t; x1, x2, x3; ξ1, ξ2, ξ3) , (19.2)

whereξk = xk − ak. (19.3)

Let us suppose that for any numerical value of the time coordi-nate in the time interval we are considering, it is possible to find

2.19 Differential rotations of a space 103

such a coordinate system (of this reference frame), at the point(19.1) of which all the functions (19.2) and their first derivativewith respect to t and also all the xk are continuous with respect toall the ξl. In this coordinate frame we have∗

(∂ ∗Δij

∂xk

)

0

=∂(∗Δij

)0

∂xk, (19.4)

(∂ ∗Δij

∂t

)

0

=∂(∗Δij

)0

∂t, (19.5)

and hence ( ∗∂ ∗Δij

∂xk

)

0

=∗∂(∗Δij

)0

∂xk, (19.6)

At the same time, the definition of the chr.inv.-tensor Dpq gives

(∗Δpq)0 = Dpq (19.7)

and hence (∗Δij

)0= Dij . (19.8)

Therefore we have( ∗∂ ∗Δij

∂xk

)

0

=∂Dij

∂xk. (19.9)

Because of (19.9) and (19.8), we have(∗∇k

∗Δij)0= ∗∇kD

ij . (19.10)

Equality (19.10), owing to its chr.inv.-tensor nature, is valid inall coordinate frames of the reference frame, not only in that speci-fic coordinate system we have been considering from the beginningof this section. Since the point (19.10)is arbitrary, equality (19.10)holds for all points (under the continuity conditions we have for-mulated above). The equality (19.10) gives

(∗∇k∗Δ)0 =

∗∇kD , (19.11)(∗∇j

∗Δij)0= ∗∇j D

ij , (19.12)

and therefore we have[∗∇j


)]

0= ∗∇j

(hijD −Dij

). (19.13)

∗The subscript zero indicates that all the ξk equal zero.


Let us again consider the neighbourhood of any point (19.1).We assume that in an arbitrary point in the neighbourhood, ∗uj

is the chr.inv.-velocity of the space with respect to the locally-stationary system at the given point (19.1). As found before, ∗Δij

is the chr.inv.-tensor of the deformation rates of the space withrespect to the aforementioned locally-stationary system, ∗ωk is thechr.inv.-vector of the angular velocities of the space rotations withrespect to the same system and ∗aij is the antisymmetric chr.inv.-tensor, dual to the chr.inv.-vector ∗ωk. The quantities are related byequations (18.9) and (18.10). At the point (19.1) itself, we have

ξi ≡ 0 , (19.14)

∗ui ≡ 0 (19.15)

and hence (∗∇j

∗aij)0=[∗∇j


)]

0, (19.16)

Ri (∗ω) =[∗∇j


)]

0. (19.17)

Because of (19.13), we can finally write(∗∇j

∗aij)0= ∗∇j

(hijD −Dij

), (19.18)

Ri (∗ω) = ∗∇j(hijD −Dij

). (19.19)

It is apparent from this algebra that the fact that the locally-stationary reference frame at the point is not unique, does notaffect our conclusions∗. This result also holds in the case where thelocally-stationary system does not rotate at this point, with respectto the space. In this case, at the point, we have

∗ωk = 0 , (19.20)

∗aij = 0 . (19.21)

§2.20 The system of locally independent quantities

Considering the problem of the space curvature, we will need touse a number of suppositions. We will give the suppositions here ina more general form than needed for solving the aforementioned

∗As mentioned in §2.11, the locally-stationary system is defined in order of itsarbitrary rotation near the point.

2.20 The system of locally independent quantities 105

problem. In accordance with §2.4, at any given world-point ofany reference frame, where the spatial coordinate frame is fixed,the potentials can be set to any preassigned numerical values byappropriate transformations of the time coordinate∗

It easy to see that we cannot set the first derivatives of thepotentials in this way, because the 16 derivatives are related by6 conditions, such that the power quantities are invariant withrespect to transformations of the time coordinate

Fi = Fi , (20.1)

Aik = Aik . (20.2)

It is evident that we can superimpose only 10 additional condi-tions on the 16 derivatives of the potentials. For this reason weintroduce 10 chr.inv.-quantities, which, being dependent on thepotentials and their first derivatives, can take any numerical valuesat any given world-point as a result of the transformations of thetime coordinate. Since we have interest in the first derivatives ofthe potentials, we can write

x0 =1

B

(Bαξ

α +1

2Bβγ ξ

βξγ)

xi = xi

, (20.3)

where B, Bα, and Bβγ are constants and B0=1,

ξα = xα − aα, (20.4)

while coordinates of the world-point are

xα = aα. (20.5)

It is evident that in general,

∂x0

∂xσ=BσB+BβσB

ξβ ,∂2x0

∂xμ∂xν=BμνB

, (20.6)

so at the world-point we have(∂x0

∂xσ

)

a

=BσB,

(∂2x0

∂xμ∂xν

)

a

=BμνB

. (20.7)

∗In this consideration, we mean that the scalar potential w cannot takenumerical values greater than or equal to c2.


Because

g00 = g00

(∂x0

∂x0

)2, g0i =

(

g00∂x0

∂xi+ g0i

)∂x0

∂x0, (20.8)

we have

c2 − w = Bc2−w

1+B0β ξβ, vi = vi +

1

c

(c2−w

)Bi+Biγ ξγ

1+B0β ξβ, (20.9)

(c2 − w

)a= B

(c2 − w

)a, (vi)a = (vi)a +

Bic

(c2 − w

)a. (20.10)

Next, let us consider the chr.inv.-derivatives and the chr.inv.-covariant derivatives of the potentials. First, this is

∗∂w

∂x0=

∗∂w

∂x0=

B

1 +B0β ξβc2

c2 − w∂w

∂x0+ c2

BB00

(1 +B0β ξβ)2 . (20.11)

At the world-point, eliminating B, we find

c2

(c2 − w)2a

( ∗∂w

∂x0

)

a

=c2

(c2 − w)2a

( ∗∂w

∂x0

)

a

+c2

(c2 − w)aB00 . (20.12)

We denote

Y =c2

(c2 − w)2∂w

∂x0, (20.13)

and then we can write

(Y )a = (Y )a +c2

(c2 − w)2aB00 . (20.14)

Hence,

∗∂w

∂xi=

∗∂w

∂xi=∂w

∂xi+

cvic2 − w

∂w

∂x0=

B

1 +B0β ξβ∂w

∂xi+

+(c2 − w)BB0i(1 +B0β ξβ)

2 +cvi

c2 − wB

(1 +B0β ξβ)

∂w

∂x0+

+cviBB00

(1 +B0β ξβ)2 =

B

1 +B0β ξβ

(∂w

∂xi+

cvic2 − w

∂w

∂x0

)

+

+(c2 − w)B

(1 +B0β ξβ)2

(

B0i +cvi

c2 − wB00

)

.

(20.15)


Because it is evident that

c2

c2 − w

(∂w

∂xi+

cvic2 − w

∂w

∂x0

)

= Fi+c2

c2 − wc ∂vi∂x0

+cviY

c2

c2 − w

(∂w

∂xi+

cvic2 − w

∂w

∂x0

)

= Fi+c2


+cviY

, (20.16)

we obtain(

Fi +c2

c2−wc∂vi∂x0

+ cviY

)

a

=

=

(

Fi +c2

c2−wc∂vi∂x0

+ cviY

)

+ c2(

cvic2−w

)

a

B00 + c2B0i .

(20.17)

On the other hand

∗∂vi∂x0

=∗∂vi∂x0

=c2

c2−w∂vi∂x0

=c2

c2−w∂vi∂x0

−c

c2−w∂w

∂x0×

×Bi +Biγ ξ

γ

1 +B0β ξβ+ c

(1 +B0β ξβ)B0i −B00(Bi +Biγ ξγ)

(1 +B0β ξβ)2

.

(20.18)

Because it is evident that

c2


= −Fi +c2

c2 − w∂w

∂xi

c2


= −Fi +c2

c2 − w∂w

∂xi

, (20.19)

we obtain(

−Fi +c2

c2 − w∂w

∂xi

)

a

=

(

−Fi +c2

c2 − w∂w

∂xi

)

a

−

−

[(c2

c2 − w∂w

∂x0

)

a

+ c2B00

]

Bi + c2B0i

(20.20)

and, taking (20.10), (20.13), and (20.14) into account, we arrive at

(

−Fi +c2

c2−w∂w

∂xi+ cviY

)

a

=

=

(

−Fi+c2

c2−w∂w

∂xi+cviY

)

a

+ c2(

cvic2−w

)

a

B00 + c2B0i .

(20.21)


Summarizing (20.17) and (20.21) term-by-term, we obtain[

c2

c2 − w

(∂w

∂xi+ c

∂vi∂x0

)

+ 2cviY

]

a

=

[c2

c2 − w×

×

(∂w

∂xi+c

∂vi∂x0

)

+ 2cviY

]

a

+2c2[(

cvic2−w

)

a

B00+B0i

]

.

(20.22)

We denote

Φi =c2

c2 − w

(∂w

∂xi+ c

∂vi∂x0

)

+ 2cviY , (20.23)

then(Φi)a=(Φi)a+ 2c2

[(cvi

c2 − w

)

a

B00 +B0i

]

. (20.24)

Taking the chr.inv.-covariant derivative of the vector potential,we obtain

∗∇i vk =∗∇i vk =

∂vk∂xi

+cvi

c2 − w∂vk∂x0

− ∗Δlik vl =

=∂vk∂xi

−1

c

∂w

∂xiBk +Bkγ ξ

γ

1 +B0β ξβ+

+c2 − wc

(1 +B0β ξβ)Bik −B0i(Bk +Bkγ ξγ)

(1 +B0β ξβ)2 +

+cvi

c2 − w∂vk∂x0

−vi

c2 − w∂w

∂x0Bk +Bkγ ξ

γ

1 +B0β ξβ+

+ vi(1 +B0β ξ

β)B0k −B00(Bk +Bkγ ξγ)

(1 +B0β ξβ)2 −

− ∗Δlikvl −1

c

(c2 − w

)∗Δlik

Bl +Blγ ξγ

1 +B0β ξβ,

(20.25)

and, on the other hand(∂vk∂xi

+cvi

c2 − w∂vk∂x0

− ∗Δlik vl

)

a

=

=

(∂vk∂xi

+cvi

c2 − w∂vk∂x0

− ∗Δlikvl

)

a

−(c2 − w)a

c2×

×

[c2

(c2 − w)a

(∂w

∂xi+

cvic2 − w

∂w

∂x0

)

a

+ c2(

cvic2 − w

)

a

B00 +

+ c2B0i

]Bkc+ (vi)aB0k +

(c2 − w)ac

[Bik −

(∗Δlik

)aBl

].

(20.26)


Owing to the symmetries on both sides[1

2

(∂vk∂xi

+∂vi∂xk

)

+1

2c2c2

c2 − w

(

vic ∂vk∂x0

+ vkc ∂vi∂x0

)

−

− ∗Δlik vl

]

a

=

[1

2

(∂vk∂xi

+∂vi∂xk

)

+1

2c2c2

c2 − w×

×

(

vic ∂vk∂x0

+ vkc ∂vi∂x0

)

− ∗Δlik vl

]

a

−(c2 − w)a2c2

×

×

{[c2

(c2−w)a

(∂w

∂xi+

cvic2−w

∂w

∂x0

)

a

+ c2(

cvic2−w

)

a

B00+

+ c2B0i

]Bkc+

[c2

(c2 − w)a

(∂w

∂xk+

cvkc2 − w

∂w

∂x0

)

a

+

+ c2(

cvkc2 − w

)

a

B00 + c2B0k

]Bic

}

+1

2

[(vi)aB0k+

+ (vk)aB0i

]+1

c

(c2 − w

)a

[Bik −

(∗Δlik

)aBl

].

(20.27)

Because of (20.10), (20.16), and (20.17), we have

−(c2 − w)a2c2

[c2

(c2 − w)a

(∂w

∂xi+

cvic2 − w

∂w

∂x0

)

a

+

+ c2(

cvic2 − w

)

a

B00 + c2B0i

]Bkc=

= −1

2c2(vk)a

(

Fi +c2


+ c viY

)

a

+

+1

2c2(vk)a

(

Fi +c2


+ c viY

)

a

+

+1

2

(cvivkc2 − w

)

a

B00 +1

2(vk)aB0i ,

(20.28)

and so obtain{1

2

(∂vk∂xi

+∂vi∂xk

)

+1

2c2

[

vk(Φi − c viY

)+

+ vi(Φk − c vkY

)]

− ∗Δlik vl

}

a

=

{1

2

(∂vk∂xi

+∂vi∂xk

)

+


+1

2c2

[vk(Φi − c viY

)+ vi

(Φk − c vkY

)]− ∗Δlikvl

}

a

+

+

(cvivkc2 − w

)

a

B00 + (vk)aB0i + (vi)aB0k +

+(c2 − w)a

c

[Bik −

(∗Δlik

)aBl

],

(20.29)

or, in the alternative form[1

2

(∂vk∂xi

+∂vi∂xk

)

+1

2c2(vkΦi + viΦk

)−

−1

cvivkY −

∗Δlik vl

]

a

=

[1

2

(∂vk∂xi

+∂vi∂xk

)

+

+1

2c2(vkΦi + viΦk

)−1

cvivkY −

∗Δlik vl

]

a

+

+

(cvivkc2 − w

)

a

B00 + (vk)aB0i + (vi)aB0k +

+(c2 − w)a

c

[Bik −

(∗Δlik

)aBl

].

(20.30)

Denoting

Xik =1

2

(∂vk∂xi

+∂vi∂xk

)

+1

2c2(viΦk + vkΦi

)−

−1

cvivkY −

∗Δlik vl ,

(20.31)

we can write

(Xik

)a=(Xik

)a+

(cvivkc2 − w

)

a

B00 + (vk)aB0i+

+ (vi)aB0k +1

c(c2 − w)a

[Bik −

(∗Δlik

)aBl

].

(20.32)

No numerical values of w, vi, Y , Φi, and Xik at the world-point can prevent selection of the unique set of the coefficientsB, Bi, Bβγ giving the quantities w, vi, Y , Φi, Xik any preassignednumerical values. Indeed, let us assume that w has a numericalvalue. Then the first of the equalities (20.10) gives B. Assuming vi,we obtain Bi from the second of the equalities (20.10). AssumingY , we obtain B00 from (20.14). Assuming Φi, we obtain B0i from

2.21 The chr.inv.-tensors of the space curvature 111

(20.24) under any numerical value of B00. Assuming Xik, we obtainBik from (20.32) for any numerical values of B00, B0l, and Bl. Thus,any of our 14 quantities can be set to any numerical value at anygiven world-point independently of the numerical values of theother quantities. Therefore, we call the 14 quantities the systemof locally-independent quantities.

It is easy to see that the quantity Y is sub-invariant, the 3 quan-tities Φi constitute a sub-vector and the 6 quantities Xik constitutea symmetric sub-tensor of the 2nd rank.

§2.21 The chr.inv.-tensors of the space curvature

Let us select that part of the chr.inv.-Riemann-Christoffel tensor,which is antisymmetric with respect to its inner pair of indices,antisymmetric with respect to the outer pair of indices, and alsosymmetric with respect to rearrangements of the outer and innerpairs of indices. First, we have

Hkjin =1

2(Hkjin −Hnjik) +

1

2(Hkjin +Hnjik) , (21.1)

where (see 15.12)1

2(Hkjin +Hnjik) =

2

c2AjiDkn , (21.2)

and also

1

2(Hkjin −Hnjik) =

1

2


−∗∂ ∗Δin,k∂xj

−


+∗∂ ∗Δjn,k∂xi

)


∗Δjk,l∗Δlin .

(21.3)

Because

1

2


−∗∂ ∗Δin,k∂xj


+∗∂ ∗Δjn,k∂xi

)

=

=1

4

( ∗∂2hkn∂xj∂xi

+∗∂2hin∂xj∂xk

−∗∂2hik∂xj∂xn

−∗∂2hnk∂xj∂xi

−


+∗∂2hin∂xj∂xk

−∗∂2hkn∂xi∂xj

−∗∂2hjn∂xi∂xk

+

+∗∂2hjk∂xi∂xn

+∗∂2hnk∂xi∂xj



)

=

=1

2

( ∗∂2hin∂xj∂xk




)

,

(21.4)


we have

1

2(Hkjin −Hnjik) =

1

2



−



)


∗Δjk,l∗Δlin .

(21.5)

Taking that part of the Riemann-Christoffel chr.inv.-Riemann-Christoffel tensor, which is antisymmetric with respect to its innerindices and outer pairs, we select that part which is symmetricwith respect to rearrangements of the outer and inner pairs of theindices. As a result, we have

1

2

(Hkjin−Hnjik

)=1

2

[1

2

(Hkjin−Hnjik

)+1

2

(Hjkni−Hiknj

)]

+

+1

2

[1

2(Hkjin−Hnjik) −

1

2(Hjkni−Hiknj)

]

,

(21.6)

and it is easy to see,

1

2

[1

2(Hkjin −Hnjik)−

1

2(Hjkni −Hiknj)

]

=

=1

4





−

−∗∂2hni∂xk∂xj

−∗∂2hkj∂xn∂xi

+∗∂2hnj∂xk∂xi

+∗∂2hki∂xn∂xj

)

=

=1

c2(AjkDin + AinDjk − AikDjn − AjnDik) ,

(21.7)

1

2

[1

2(Hkjin −Hnjik) +

1

2(Hjkni −Hiknj)

]

=

=1

4


+∗∂2hin∂xk∂xj

)

+1

4

( ∗∂2hjk∂xi∂xn

+∗∂2hjk∂xn∂xi

)

−

−1

4

( ∗∂2hik∂xj∂xn

+∗∂2hik∂xn∂xj

)

−1

4

( ∗∂2hjn∂xi∂xk

+∗∂2hjn∂xk∂xi

)

−


∗Δjk,l∗Δlin .

(21.8)

Denoting that part of the chr.inv.-Riemann-Christoffel tensor


we have selected by Skjin

Skjin =1

4



)

+

+1

4



)

−1

4



)

−

−1

4



)

+ ∗Δlin∗Δjk,l −

∗Δlik∗Δjn,l ,

(21.9)

we can write

Hkjin = Skjin +1

c2(2AjiDkn + AinDjk+

+ AnjDik + AjkDin + AkiDjn).

(21.10)

Because Hkjin, Apq, and Dpq are chr.inv.-tensors, Skjin is also achr.inv.-tensor. We will refer to Skjin as the chr.inv.-curvature ten-sor of the 4th rank. Thus, the chr.inv.-Riemann-Christoffel tensordiffers from the curvature chr.inv.-tensor of the 4th rank, whilethe Riemann-Christoffel co-tensor and the curvature co-tensor ofthe 4th rank are merely different names for the same sub-tensor, aformula for which can be written in the form

Kkjin =1

2

(∂2hin∂xj∂xk

+∂2hjk∂xi∂xn

−∂2hik∂xj∂xn

−∂2hjn∂xi∂xk

)

+

+ ΔlinΔjk,l −ΔlikΔjn,l ,

(21.11)

equivalent to (15.9) and like (21.9). Defining the quantity Skjin as

Skjin = −Skijn , (21.12)

Skjin = −Snjik , (21.13)

Skjin = Sjkni , (21.14)

we can see that the equalities are true, because of (21.9).We also introduce the chr.inv.-curvature tensor of the 2nd rank

(which is different from the chr.inv.-Einstein tensor)

Skj = Skjinhin = S∙∙∙lkjl . (21.15)

Formula (21.14) leads to that

Skj = Sjk . (21.16)


Because

Skjin =1

2

[1

2(Hkjin −Hnjik) +

1

2(Hjkni −Hiknj)

]

, (21.17)

and taking (21.2) into account, we have

Skjin =1

2(Hkjin +Hjkni)−

1

c2(AjiDkn + AknDji) , (21.18)

Skj =1

2(Hkj +Hjk)−

1

c2(AjlD

lk + AklD

lj

). (21.19)

Owing to (15.21), we have

Skj = Hkj −1

c2(AjkD + AjlD

lk + AklD

lj

), (21.20)

Hkj = Skj +1

c2(AjkD + AjlD

lk + AklD

lj

), (21.21)

that can also be obtained directly from (21.10).We also introduce the chr.inv.-curvature invariant

S = hkjSkj . (21.22)

Formula (21.19) givesS = H. (21.23)

Because of the symmetric properties (21.12–21.14), which areanalogous to the symmetric properties of the curvature co-tensorof the 4th rank, the number of the different components of Skjinequals the number of the different components of Kkjin, i. e. thereare only 6 different components. Therefore, by analogy with Ricci’scontravariant and covariant tensors (see [8], p. 110)

Cab =1

4εaijεbknKkjin , (21.24)

Crs = hrahsbCab, (21.25)

we can introduce the chr.inv.-tensors

Zab =1

4εaijεbknSkjin , (21.26)

Zrs = hrahsbZab, (21.27)


which will be referred to as the chr.inv.-Ricci tensors. The Riccico-tensor is linked to the curvature co-tensor of the 2nd rank andthe curvature co-invariant by the equation

Crq = Krq −1

2hrqK . (21.28)

The same relation holds between the chr.inv.-Ricci tensor, thechr.inv.-curvature tensor of the 2nd rank, and the chr.inv.-curva-ture invariant

Zrq = Srq −1

2hrqS . (21.29)

We have

εapq εbrsZab =

1

4εaijεapq ε

bknεbrsSkjin =

=1

4

(hiph

jq − h

iqh

jp

)(hkrh

ns − h

ksh

nr

)Skjin =

=1

4

(hiph

jq − h

iqh

jp

)(Srjis − Ssjir) =

=1

2

(hiph

jq − h

iqh

jp

)Srjis =

1

2(Srqps − Srpqs) = Srqps ,

(21.30)

henceSrq = hpsεapq εbrsZ

ab. (21.31)


hps =1

2εijpεklshikhjl , (21.32)

hence

Srq =1

2εijpεapq ε

klsεbrs hikhjlZab =

= −1

2

(hiah

jq − h

iqh

ja

)(hkb h

lr − h

krh

lb

)hikhjlZ

ab =

= −1

2(hakhql−hkqhal)

(hlrZ

ak−hkrZal)= Zrq−hrqZ ,

(21.33)

Z = hikZik , (21.34)

S = −2Z . (21.35)

Formulae (21.33) and (21.35) lead to (21.29), which is theirconsequence.


§2.22 On the curvature of space

We are going to find the relation between the chr.inv.-curvaturetensor of the 4th rank and the curvature co-tensor of the 4th rank.First, we have

∗∂2hin∂xj∂xk

=∗∂

∂xj

(∂hin∂xk

+2

c2vkDin

)

=∂2hin∂xj∂xk

+

+vj

c2 − w∂2hin∂t ∂xk

+2

c2

(∂vk∂xj

+vj

c2 − w∂vk∂t

)

Din +

+2

c2vk

∗∂Din∂xj

=∂2hin∂xj∂xk

−2

c2vj

c2 − w∂w

∂xkDin +

+2

c2vj

(∗∂Din∂xk

−vkc2

∗∂Din∂t

)

+2

c2

(∂vk∂xj

+vj

c2−w∂vk∂t

)

Din+

+2

c2vk

∗∂Din∂xj

=∂2hin∂xj∂xk

+2

c2

(∂vk∂xj

−1

c2Fkvj

)

Din +

+2

c2

(

vj∗∂Din

∂xk+ vk

∗∂Din∂xj

)

−2

c4vjvk

∗∂Din

∂t,

(22.1)

and also

1

2



)

=∂2hin∂xj∂xk

+2

c2ΨjkDin+

+2

c2( vj

∗∇kDin + vk∗∇jDin)−

2

c4vjvk

∗∂Din

∂t+

+2

c2(∗ΔljkDin vl +

∗ΔlkiDnl vj +∗ΔlknDil vj +

+ ∗ΔljiDnl vk +∗ΔljnDilvk

),

(22.2)

where we denote

Ψjk =1

2

(∂vk∂xj

+∂vj∂xk

)

−1

2c2(Fk vj + Fj vk)−

∗Δljk vl . (22.3)

Therefore, we have

1

2

(∂2hin∂xj∂xk

+∂2hjk∂xi∂xn

−∂2hik∂xj∂xn

−∂2hjn∂xi∂xk

)

=

=1

4



)

+1

4



)

−

2.22 On the curvature of space 117

−1

4



)

−1

4



)

−

−1

c2(ΨjkDin +ΨinDjk −ΨjnDik −ΨikDjn)−

−1

c2( vj

∗∇kDin + vk∗∇jDin + vi

∗∇nDjk + vn∗∇iDjk−

− vj∗∇nDik − vn

∗∇jDik − vi∗∇kDjn − vk

∗∇iDjn) +

+1

c4

(

vjvk∗∂Din

∂t+vivn

∗∂Djk∂t

−vjvn∗∂Dik

∂t−vivk

∗∂Djn

∂t

)

−

−1

c2(∗ΔljkDinvl+

∗ΔlkiDnl vj+∗ΔlknDil vj+

∗ΔljiDnl vk+

+ ∗ΔljnDilvk +∗ΔlinDjk vl +

∗ΔlnjDkl vi +∗ΔlnkDjl vi +

+ ∗ΔlijDkl vn +∗ΔlikDjl vn −

∗ΔljnDik vl −∗ΔlniDkl vj −

− ∗ΔlnkDil vj −∗ΔljiDkl vn −

∗ΔljkDil vn −∗ΔlikDjn vl −

− ∗ΔlkjDnl vi −∗ΔlknDjl vi −

∗ΔlijDnl vk −∗ΔlinDjl vk

)=

=1

4



)

+1

4



)

−

−1

4



)

−1

4



)

−

−1

c2(ΨjkDin +ΨinDjk −ΨjnDik −ΨikDjn)−

−1

c2

[vj (

∗∇kDin−∗∇nDik) + vk (

∗∇jDin−∗∇iDjn) +

+ vi (∗∇nDjk −

∗∇kDjn) + vn (∗∇iDjk −

∗∇jDik)]+

+1

c4

(

vjvk∗∂Din

∂t+vivn

∗∂Djk∂t

−vjvn∗∂Dik

∂t−vivk

∗∂Djn

∂t

)

+

+1

c2

[∗Δljk (Dil vn +Dnl vi −Din vl) +

∗Δlin (Djl vk +

+ Dkl vj −Djk vl)−∗Δljn (Dilvk +Dkl vi −Dik vl)−

− ∗Δlik (Djl vn +Dnl vj −Djn vl)].

(22.4)

We then obtain


ΔlinΔjk,l−ΔlikΔjn,l =

[∗Δlin−

1

c2(Dlivn+D

lnvi−Dinv

l)]×

×[∗Δjk,l −

1

c2(Djl vk +Dkl vj − Djk vl

)]−

−[∗Δlik −

1

c2(Dlivk +D

lk vi −Dik v

l)]×

×[∗Δjn,l −

1

c2(Djlvn +Dnl vj −Djnvl

)]= ∗Δlin

∗Δjk,l −

− ∗Δlik∗Δjn,l −

1

c2∗Δjk,l

(Dlivn +D

lnvi −Din v

l)−

−1

c2∗Δlin

(Djl vk +Dkl vj − Djk vl

)+1

c2∗Δjn, l

(Dlivk+

+ Dlk vi −Dik v

l)+1

c2∗Δlik

(Djl vn +Dnlvj −Djn v

l)+

+1

c4

[(Djl vk +Dkl vj −Djk vl

)(Dlivn +D

lnvi −Dinv

l)−

−(Djl vn +Dnl vj − Djn vl

)(Dlivk +D

lk vi −Dik v

l)],

(22.5)

hence,

Kkjin = Skjin −1

c2(ΨjkDin +ΨinDjk−

− ΨjnDik −ΨinDjk)−1

c2

[vj (

∗∇kDin −∗∇nDik) +

+ vk (∗∇jDin −

∗∇iDjn) + vi (∗∇nDjk −

∗∇kDjn) +

+ vn (∗∇iDjk −

∗∇jDik)]+1

c4

(

vjvk∗∂Din

∂t+

+ vivn∗∂Djk∂t

− vjvn∗∂Dik∂t

− vivk∗∂Djn

∂t

)

+

+1

c4(− vjvkDilD

ln − vivnDjlD

lk + vjvnDilD

lk +

+ vivkDjlDln

)+1

c4

[− vl

(Djl vk +Dkl vj −Djk vl

)Din −

− vl(Dlj vn +D

ln vj −Djn v

l)Dik + v

l(Dil vn + Dnl vi −

− Dinvl)Djk + vl

(Dlivk +D

lk vi −Dik v

l)Djn

]+

+1

c4vlvl

(−DjkDin +DjnDik

).

(22.6)


Introducing

Σjk =1

2

(∂vk∂xj

+∂vj∂xk

)

−1

2c2(Fk vj + Fj vk)−Δ

ljk vl (22.7)

instead of Ψjk , i. e.

Ψjk = Σjk −1

c2(Dlj vk +D

lk vj −Djk v

l)vl , (22.8)

and solving equation (22.6) with respect to Skjin, we can finallywrite

Skjin = Kkjin +1

c2

[ΣjkDin +ΣinDjk − ΣjnDik−

− ΣikDjn]+1

c4vlv

l[DjkDin −DjnDik

]+

+1

c4

[

vjvk

(

DilDln−

∗∂Din

∂t

)

+ vivn

(

DjlDlk−

∗∂Djk

∂t

)

−

− vjvn

(

DilDlk −

∗∂Dik

∂t

)

− vivk

(

DjlDln −

∗∂Djn

∂t

)]

+

+1

c2

[vk (

∗∇jDin −∗∇iDjn) + vj (

∗∇kDin −∗∇nDik)+

+ vi (∗∇nDjk −

∗∇kDjn) + vn (∗∇iDjk −

∗∇jDik)].

(22.9)

Comparing formulae (22.7) and (20.31), we verify that the con-ditions

vi = 0

Σjk = 0

}

(22.10)

are equivalent to the conditions

vi = 0

Xjk = 0

}

. (22.11)

Hence, there are ∞5 systems of the numerical values of thecoefficients B, Bi, Bβγ in (20.3) which, at the world-point we areconsidering, permit the conditions (22.10) and also the equality

Skjin = Kkjin . (22.12)


Note that the conditions (22.10) or (22.11) are equivalent to theconditions

g0i = 0

∂g0k∂xj

+∂g0j∂xk

= 0

. (22.13)

We now have a possibility of finding the geometrical definitionof “the space curvature” (see the geometrical definition of space atthe beginning of §2.2). We assume a world-point

xσ = aσ, σ = 0, 1, 2, 3 , (22.14)

at which we take the spatial section

x0 = a0 (22.15)

of the four-dimensional world. We define the metric in this spatialsection by the sub-tensor


. (22.16)

It is evident that the curvature of this spatial section will bedefined by the sub-tensor Kkjin. Transforming the ime coordinate,we obtain another spatial section, fixed through the same world-point. In the new spatial section, the numerical values of hik at theworld-point will be the same, while components of the curvaturesub-tensor Kkjin will have, generally speaking, different numericalvalues. We will limit the circle of the spatial sections we are con-sidering. We will consider only those spatial sections which are or-thogonal (at this world-point) to the time lines of this reference fra-me. In other words, we will consider those spatial sections where,at the world-point,

g0i = 0 . (22.17)

In this case we havevi = 0 , (22.18)

Skjin = Kkjin +1

c2

[ΣjkDin + ΣinDjk − ΣjnDik − ΣikDjn

]. (22.19)

Since by (22.18) we have

Σjk = Xjk , (22.20)


hence, at this world-point,

Skjin = Kkjin +1

c2

[XjkDin+XinDjk−XjnDik−XikDjn

]. (22.21)

It is also evident that

S = K +2

c2

[XD − X l

jDjl

], (22.22)

whereX = Xikh

ik. (22.23)

Changing values of Xik, we change values of Kkjin and K aswell. Hence, different spatial sections, which are orthogonal to timelines at the given world-point, have, generally speaking, differentcurvatures at the point. Moreover, both their Riemannian curvatu-res and their scalar (mean) curvatures are different. Now, we willconsider only those spatial sections, where the conditions (22.10)are true at the world-point. We will refer to them as the maximallyorthogonal spatial sections. In all the spatial sections, the conditions(22.10) and (22.12) are true. Therefore their curvatures (both theRiemannian curvature and the mean curvature) are the same.

Taking “space” in the sense of §2.2, we can give the geometricdefinition of its curvature∗ as follows:

We will mean by the “curvature” of the space in a given world-point the curvature of any spatial section which is maximallyorthogonal at this world-point.

This definition of the space curvature justifies the terminology wegave, in accordance with which the chr.inv.-tensors Skjin and Skjare the chr.inv.-curvature tensors, the chr.inv.-invariant S is thechr.inv.-curvature invariant, and Zkj is the chr.inv.-Ricci tensor.

Next we assume that U j is the chr.inv.-unit vector, which isorthogonal (at the given world-point) to the two-dimensional di-rection we are considering. Having the spatial section in the world-point fixed, we find that its Riemannian curvature CR (U) along theaforementioned direction is

CR (U) = Crq UrUq = Krq U

rUq −1

2K , (22.24)

and because of (21.28),

hrq UrUq = 1 . (22.25)

∗The Riemannian curvature — along the two-dimensional direction, the scalarcurvature — on the average.


Calculating the mean curvature CN of the spatial section in thisworld-point, we obtain

CN =1

3C = −

1

6K , (22.26)

whereC = hikCik . (22.27)

So, because of what has been said above, we can write theRiemannian curvature ∗CR of the space in the given world-point(along the two-dimensional direction we are considering) and alsoits mean curvature CN in the point as follows

∗CR (U) = Zrq UrUq = Srq U

rUq −1

2S (22.28)

and∗CN =

1

3Z = −

1

6S . (22.29)

♦

Chapter 3

RELATIVISTIC PHYSICAL EQUATIONS

§3.1 The metric tensor

The main task of this section is to put the equations of the Gen-eral Theory of Relativity into chronometrically invariant (chr.inv.-tensor) form. We will first consider the metric world-tensor.

In Chapter 2, we have obtained the covariant metric tensorequations (2.9), (2.10), and (8.9)

g00 =(1−

w

c2

)2, (1.1)

g0i = −(1−

w

c2

) vic, (1.2)

gik = −hik +vivkc2

, (1.3)

the contravariant metric tensor equations (8.25), (8.24), (8.14)

g00 =1

(1− w

c2

)2

(

1−vjv

j

c2

)

, (1.4)

g0i = −1

1− wc2

vi

c, (1.5)

gik = −hik, (1.6)

and the fundamental determinant g, equation (8.19)

√−g =

(1−

w

c2

)√h . (1.7)

124 Chapter 3 Relativistic Physical Equations

§3.2 Christoffel’s symbols of the 1st kind

Because∂g00∂x0

= −2

c2

(1−

w

c2

) ∂w∂x0

, (2.1)

we have

Γ00,0 = −1

c3

(1−

w

c2

) ∂w∂t

. (2.2)

and so we obtain

∂g00∂xi

= −2

c2

(1−

w

c2

) ∂w∂xi

, (2.3)

∂g0i∂x0

=1

c3vi∂w

∂x0−1

c

(1−

w

c2

) ∂vi∂x0

(2.4)

and also

∂g0i∂x0

−1

2

∂g00∂xi

=1

c3vi∂w

∂x0+1

c2

(1−

w

c2

)( ∂w∂xi

− c∂vi∂x0

)

=

=1

c2

(1−

w

c2

)2Fi +

1

c4vi∂w

∂t,

(2.5)

because of formula (6.5) of §2.6. Thus, we arrive at

Γ00,i =1

c2

(1−

w

c2

)2Fi +

1

c4vi∂w

∂t. (2.6)

Because∂g00∂xi

= −2

c2

(1−

w

c2

) ∂w∂xi

, (2.7)

we have

Γ0i,0 = −1

c2

(1−

w

c2

) ∂w∂xi

. (2.8)

and so we obtain

∂g0j∂xi

=1

c3vj∂w

∂xi−1

c

(1−

w

c2

)∂vj∂xi

, (2.9)

∂gij∂x0

= −2

c2

(1−

w

c2

)Dij +

1

c2

(

vj∂vi∂x0

+ vi∂vj∂x0

)

(2.10)

3.2 Christoffel’s symbols of the 1st kind 125

and, hence

1

2

(∂gij∂x0

+∂g0j∂xi

−∂g0i∂xj

)

= −1

c

(1−

w

c2

)×

×

[

Dij +1

2

(∂vj∂xi

−∂vi∂xj

)]

+1

2c2vj∂vi∂x0

−1

2c3vj∂w

∂xi+

+1

c3vj∂w

∂xi−

1

2c2vi∂vj∂x0

+1

c2vi∂vj∂x0

+1

2c3vi∂w

∂xj−

−1

c3vi∂w

∂xj= −

1

c

(1−

w

c2

)[

Dij +1

2

(∂vj∂xi

−∂vi∂xj

)

+

+1

2c2(Fivj − Fjvi) +

1

c2Fjvi

]

+1

c3vj∂w

∂xi=

= −1

c

(1−

w

c2

)(Dij + Aij +

1

c2Fjvi

)+1

c3vj∂w

∂xi.

(2.11)

Thus, we arrive at

Γ0i,j = −1

c

(1−

w

c2

)(Dij + Aij +

1

c2Fjvi

)+1

c3vj∂w

∂xi. (2.12)

Because of (2.9) and (2.10), we have

1

2

(∂g0j∂xi

+∂g0i∂xj

−∂gij∂x0

)

=1

2c3

(

vj∂w

∂xi+ vi

∂w

∂xj

)

−

−1

2c

(1−

w

c2

)(∂vj∂xi

+∂vi∂xj

)

+1

c

(1−

w

c2

)Dij −

−1

2c2

(

vj∂vi∂x0

+ vi∂vj∂x0

)

=1

c

(1−

w

c2

)×

×

[

Dij −1

2

(∂vj∂xi

+∂vi∂xj

)

+1

2c2(Fivj + Fjvi)

]

,

(2.13)

and hence

Γij,0 =1

c

(1−

w

c2

)[

Dij−1

2

(∂vj∂xi

+∂vi∂xj

)

+1

2c2(Fivj+Fjvi)

]

. (2.14)

Finally, because

∂gij∂xk

= −∂hij∂xk

+1

c2

(

vj∂vi∂xk

+ vi∂vj∂xk

)

, (2.15)


we have

1

2

(∂gjk∂xi

+∂gik∂xj

−∂gij∂xk

)

= −1

2

(∂hjk∂xi

+∂hik∂xj

−∂hij∂xk

)

+

+1

2c2

[

vi

(∂vk∂xj

−∂vj∂xk

)

+ vj

(∂vk∂xi

−∂vi∂xk

)

+

+ vk

(∂vj∂xi

+∂vi∂xj

)]

+1

2c4

[vi (Fjvk − Fkvj) +

+ vj (Fivk − Fkvi)− vk (Fivj + Fjvi) + 2Fkvivj], .

(2.16)

and so

Γij,k = −Δij,k+1

c2

[

viAjk+vjAik+1

2vk

(∂vj∂xi

+∂vi∂xj

)

−

−1

2c2vk (Fivj + Fj vi)

]

+1

c4Fk vivj .

(2.17)

Formulae (2.2), (2.6), (2.8), (2.12), (2.14), and (2.17) collected intheir final forms are

Γ00,0 = −1

c3

(1−

w

c2

) ∂w∂t

, (2.18)

Γ00,i =1

c2

(1−

w

c2

)2Fi +

1

c4vi∂w

∂t, (2.19)

Γ0i,0 = −1

c2

(1−

w

c2

) ∂w∂xi

, (2.20)

Γ0i,j = −1

c

(1−

w

c2

)(Dij + Aij +

1

c2Fj vi

)+1

c3vj∂w

∂xi, (2.21)

Γij,0 =1

c

(1−

w

c2

)[

Dij−1

2

(∂vj∂xi

+∂vi∂xj

)

+1

2c2(Fivj+Fjvi)

]

, (2.22)

Γij,k = −Δij,k+1

c2

[

viAjk+vjAik+1

2vk

(∂vj∂xi

+∂vi∂xj

)

−

−1

2c2vk(Fivj + Fjvi)

]

+1

c4Fk vivj .

(2.23)

3.3 Christoffel’s symbols of the 2nd kind 127

§3.3 Christoffel’s symbols of the 2nd kind


g00Γ00,0 + g0kΓ00,k = −

1

c3c2

c2 − w

(

1−vjv

j

c2

)∂w

∂t−

−1

c3

(1−

w

c2

)vkFk −

1

c5c2

c2 − wvjv

j ∂w

∂t=

= −1

c3

[c2

c2 − w∂w

∂t+(1−

w

c2

)vkF

k

]

,

(3.1)

hence

Γ000 = −1

c3

[c2

c2 − w∂w

∂t+(1−

w

c2

)vkF

k

]

. (3.2)

Besides these, because

gk0Γ00,0 + gklΓ00,l =

1

c4vk∂w

∂t−1

c2

(1−

w

c2

)2F k−

1

c4vk∂w

∂t, (3.3)

we get

Γk00 = −1

c2

(1−

w

c2

)2F k. (3.4)

Because of (2.20) and (2.21)

g00Γ0i,0 + g0kΓ0i,k = −

1

c2c2

c2 − w

(

1−vjv

j

c2

)∂w

∂xi+

+1

c2vk(Dik + Aik +

1

c2Fkvi

)−1

c2vkv

k

c2 − w∂w

∂xi=

= −1

c2c2

c2 − w∂w

∂xi+1

c2vk

(Dki + A

∙ki∙ +

1

c2viF

k),

(3.5)

we obtain

Γ00i =1

c2

[

−c2

c2 − w∂w

∂xi+ vk

(Dki + A

∙ki∙ +

1

c2viF

k)]

(3.6)

and also

gk0Γ0i,0+gklΓ0i,l=

1

c

(1−w

c2

)(Dki+A

∙ki∙+

1

c2viF

k), (3.7)

Γk0i =1

c

(1−

w

c2

)(Dki + A

∙ki∙ +

1

c2viF

k). (3.8)


Because of (2.22) and (2.23), we obtain

g00Γij,0 + g0kΓij,k =

1

c

c2

c2 − w

(

1−vkv

k

c2

)

×

×

[

Dij −1

2

(∂vj∂xi

+∂vi∂xj

)

+1

2c2(Fivj + Fjvi)

]

+

+1

c

c2

c2 − wΔij,k v

k −1

c3c2

c2 − wvk

{

viAjk + vjAik +

+ vk

[1

2

(∂vj∂xi

+∂vi∂xj

)

−1

2c2(Fivj + Fjvi)

]}

−

−1

c5c2

c2 − wvkFkvivj =

1

c

c2

c2 − w

(

1−vkv

k

c2

)

Dij −

−1

c

c2

c2 − wΣij −

1

c3c2

c2 − wvk (viAjk + vjAik)−

−1

c5c2

c2 − wvkFkvivj ,

(3.9)

that is

Γ0ij = −1

c

c2

c2 − w

[

Σij −

(

1−vkv

k

c2

)

Dij +

+1

c2vk(viA

∙kj∙ + vjA

∙ki∙

)+1

c4vivjvkF

k

]

,

(3.10)

where Σij is defined by formula (22.7), §2.22. Thus, we obtain

gk0Γij,0 + gklΓij, l = −

1

c2vk[

Dij −1

2

(∂vj∂xi

+∂vi∂xj

)

+

+1

2c2(Fivj + Fjvi)

]

+Δkij −1

c2

{

viA∙kj∙ + vjA

∙ki∙ +

+ vk[1

2

(∂vj∂xi

+∂vi∂xj

)

−1

2c2(Fivj+Fjvi)

]}

−1

c4F kvivj =

= Δkij −1

c2vkDij −

1

c2(viA

∙kj∙ + vjA

∙ki∙

)−1

c2F k vivj ,

(3.11)

which gives

Γkij = Δkij −

1

c2(Dij v

k + viA∙kj∙ + vjA

∙ki∙ +

1

c2vivjF

k). (3.12)

We introduce Ψij instead of Σij into formula (3.10), and also ∗Δkijinstead of Δkij into (3.12). Then, according to formula (22.8) of

3.3 Christoffel’s symbols of the 2nd kind 129

§2.22, we obtain

Σij−

(

1−vkv

k

c2

)

Dij = Ψij−

(

1−vkv

k

c2

)

Dij+1

c2(Dki vj +

+ Dkj vi −Dij v

k)vk = Ψij −Dij +

1

c2vk(Dkj vi +D

ki vj),

(3.13)

hence

Γ0ij = −1

c

c2

c2 − w

{

Ψij −Dij +1

c2vk

[(Dkj + A

∙kj∙

)vi+

+(Dki + A

∙ki∙

)vj +

1

c2vivjF

k]}

.

(3.14)

Because, in accordance with formula (13.8) of §2.13,

Δkij −1

c2Dij v

k = ∗Δkij −1

c2(Dki vj +D

kj vi), (3.15)

we get

Γkij =∗Δkij −

1

c2

[

vi(Dkj +A

∙kj∙

)+ vj

(Dki +A

∙ki∙

)+1

c2vivjF

k

]

. (3.16)

Formulae (3.2), (3.4), (3.6), (3.8), (3.14), and (3.16) collected intheir final forms are

Γ000 = −1

c3

[c2

c2 − w∂w

∂t+(1−

w

c2

)vkF

k

]

, (3.17)

Γk00 = −1

c2

(1−

w

c2

)2F k, (3.18)

Γ00i =1

c2

[

−c2

c2 − w∂w

∂xi+ vk

(Dki + A

∙ki∙ +

1

c2viF

k)]

, (3.19)

Γk0i =1

c

(1−

w

c2

)(Dki + A

∙ki∙ +

1

c2viF

k), (3.20)

Γ0ij = −1

c

c2

c2 − w

{

−Dij +1

c2vk

[vi(Dkj + A

∙kj∙

)+

+ vj(Dki + A

∙ki∙

)+1

c2vivjF

k]+1

2

(∂vj∂xi

+∂vi∂xj

)

−

−1

2c2(Fivj + Fjvi)−

∗Δkij vk

}

,

(3.21)

Γkij =∗Δkij−

1

c2

[

vi(Dkj +A

∙kj∙

)+ vj

(Dki +A

∙ki∙

)+1

c2vivjF

k

]

. (3.22)


We will apply the Christoffel symbols, when we deduce equa-tions for the dynamics of a point-mass in order to find the mech-anical sense of the quantities Fi and Ajk. However, before we do itthat, we need to consider the problem of the square of the velocityof light, and also introduce the mass, energy, and momentum of atest point-body.

§3.4 The speed of light

It easy to see that we can re-write the equality

ds2 = g00dx0dx0 + 2g0idx

0dxi + gik dxidxk (4.1)

in the form

ds2 =

(√g00 dx

0 +g0i√g00

dxi)2+

(

gik −g0ig0kg00

)

dxidxk, (4.2)


ds2 =

(dx0√g00

)2− hik dx

idxk. (4.3)

Hence, we have(√g00

ds

dx0

)2= 1− hik

(√g00

dxi

dx0

)(√g00

dxk

dx0

)

(4.4)

and, in accordance with §2.9,

√g00

dxi

dx0=

∗ui

c. (4.5)

We therefore obtain(√g00

ds

dx0

)2= 1− hik

∗ui∗uk

c2(4.6)

or, in the alternative form(√g00

ds

dx0

)2= 1−

∗uj∗uj

c2. (4.7)

For light rays we haveds = 0 (4.8)

and, hence∗uj

∗uj = c2. (4.9)

So the square of the chr.inv.-vector for the velocity of lightequals c2 in emptiness.

3.5 A point-body. Its mass, energy, and the momentum 131

§3.5 A point-body. Its mass, energy, and the momentum

Using the transformations (4.9) of §2.4, we can set the potentialsto zero at any given world-point. Then, having the potentials zeroat the point, we can write equations for the mass m, energy E,and momentum p of a point-mass analogous to the appropriateequations of the Special Theory of Relativity. Namely, we can write

m = m0dx0

ds, (5.1)

E = mc2, (5.2)

pi = m0c dxi

ds. (5.3)

At the given world-point in the coordinates xσ (σ = 0, 1, 2, 3) wehave chosen we have

g00 = 1 , g0i = 0 (5.4)

and hence

dx0 = dx0 =dx0√g00

, (5.5)

and so we obtain

m =m0√g00

dx0ds

. (5.6)

At the same time the quantity

dx0√g00

=dx0√g00

(5.7)

is a chr.inv.-invariant in any arbitrary system of the spatial coordi-nates xσ (σ=0, 1, 2, 3) of the given reference frame. For this reason,in general, we have

m =m0√g00

dx0ds

(5.8)

and, because of (4.7),

m =m0√

1−∗ui

∗ui

c2

. (5.9)

We have thus obtained equations for the chr.inv.-invariant ofmoving (relativistic) mass. The chr.inv.-invariant of energy can be


expressed by formula (5.2). So, with §3.4 as a basis, we can say thatthe energy of a point-mass equals its dynamic mass, multiplied bythe square of the velocity of light.

Because dxi is a chr.inv.-vector, i. e.

dxi = dxi, (5.10)

in the general case we have

pi = m0c dxi

ds. (5.11)

This is the equation of the chr.inv.-vector of momentum. As wehave obtained

dxi

ds=

(√g00

dxi

dx0

)(1

√g00

dx0ds

)

, (5.12)

then, taking (4.5) into account, we arrive at

pi = m∗ui. (5.13)

§3.6 Transformations of the energy and momentum of a point-body

With our results for chr.inv.-derivatives and chr.inv.-velocities abasis, we can say that the operator, invariant with respect to trans-formations of time the coordinate and also, with the potentials areset to zero, coincident with the operator of the total derivative withrespect to the time coordinate

d

dt=

∂

∂t+ uj

∂

∂xj, (6.1)

take the form∗d

dt=

∗∂

∂t+ ∗uj

∗∂

∂xj. (6.2)

We will refer to the operator as the operator of the total deriva-tive with respect to the time coordinate.

Let us find the relation between the chr.inv.-total derivativewith respect to time, on one the hand, and, on the other hand, thederivative with respect to local time and the partial derivative withrespect to time

∗d

dt=

c2

c2−w∂

∂t+

c2uj

c2−w−vkuk∂

∂xj+

vjc2−w

∗uj∂

∂t=

=c2

c2−w

(1 +

1

c2vj∗uj) ∂∂t+

c2

c2−wc2−w

c2−w−vkukuj

∂

∂xj.

(6.3)

3.6 Transformations of a point-body’s energy-momentum 133

We know that

1 +1

c2vj∗uj = 1 +

vjuj

c2 − w − vkuk=

c2 − wc2 − w − vjuj

, (6.4)

hence we have∗d

dt=

c2

c2 − w

(1 +

1

c2vj∗uj) ddt

(6.5)


∗d

dt=

1

1− 1c2(w + vjuj)

d

dt. (6.6)

Because

dx0dx0

= g00 + g0jdxj

dx0=(1−

w

c2

)(

1−w

c2−vju

j

c2

)

=

=1

1 +1c2vj ∗uj

(1−

w

c2

)2,

(6.7)

and taking formula (4.7) into account, we obtain

dx0

ds=

c2

c2 − w

1 + 1c2vj∗uj

√

1−∗ui

∗ui

c2

. (6.8)


∗d

dt=

√

1−∗uk∗uk

c2cd

ds. (6.9)

Formula (6.2) leads also to

∗d

dt=(1 +

1

c2vj∗uj) c2

c2 − w∂

∂t+ ∗uj

∂

∂xj. (6.10)

Employing the formulae we have obtained for the energy of apoint-mass, we, in particular, have

∗dE

dt=

√

1−∗uk∗uk

c2cdE

ds. (6.11)


Because E is a chr.inv.-invariant,∗dEdt

is chr.inv.-invariant aswell. The total derivative

dQk

dt=∂Qk

∂t+ uj

∂Qk

∂xj(6.12)

of any sub-vector Qk is not sub-vector. At the same time, it is pos-sible to introduce the sub-vector

dQk

dt+ΔkjlQ

luj =∂Qk

∂t+(∇jQ

k)uj , (6.13)

which characterizes the total derivative.The chr.inv.-total derivative of Qk, the quantity

∗dQk

dt=

∗∂Qk

∂t+ ∗uj

∗∂Qk

∂xj, (6.14)

is not a sub-vector either. We can introduce the operation of chr.inv.-total differentiation, which, being invariant with respect to thetransformations of the time coordinate, coincides with the regularoperation of obtaining the sub-vector of the total derivative whenthe potentials are set to zero.

The sub-vector of the chr.inv.-total derivative of Qk is differentto the regular formula (6.13) only by the presence of asterisks

∗dQk

dt+ ∗ΔkjlQ

l ∗uj =∗∂Qk

∂t+(∗∇jQ

k)∗uj . (6.15)

Using (6.4–6.6), we obtain

∗dQk

dt+ ∗ΔkjlQ

l ∗uj =c2

c2 − w

(1 +

1

c2vn∗un)×

×

[dQk

dt+ΔkjlQ

luj +1

c2(Dkj vl +D

kl vj −Djl v

k)Qluj

] (6.16)


∗dQk

dt+ ∗ΔkjlQ

l ∗uj =1

1− 1c2(w + vnun)

×

×

[dQk

dt+ΔkjlQ

luj +1

c2(Dkj vl +D

kl vj −Djl v

k)Qluj

]

.

(6.17)

3.7 The time equation of geodesic lines 135

Comparing (5.11) and (5.13), and taking formula (5.9) into ac-count, we obtain

cdxi

ds=

∗ui√

1−∗uk

∗uk

c2

. (6.18)


∗dQk

dt+ ∗ΔkjlQ

l ∗uj = c

√

1−∗un∗un

c2

(dQk

ds+ ∗ΔkjlQ

l dxj

ds

)

. (6.19)

Applying the obtained formulae to the momentum of a point-mass, and supposing that its rest-mass m0 remains unchanged, wecan, in particular, write

∗dpk

dt+∗Δkjl p

l∗uj = m0c2

√

1−∗un∗un

c2

(d2xk

ds2+∗Δkjl

dxj

ds

dxl

ds

)

. (6.20)

Because the sub-vector of momentum is chr.inv.-vector, thesub-vector of the total derivative of momentum with respect totime is also a chr.inv.-vector.

§3.7 The time equation of geodesic lines

With the four equations of geodesic lines

d2xα

ds2+ Γαμν

dxμ

ds

dxν

ds= 0 , α, μ, ν = 0, 1, 2, 3 , (7.1)

we are going to consider the time equation (α = 0)

d2x0

ds2+ Γ000

dx0

ds

dx0

ds+ 2Γ00i

dx0

ds

dxi

ds+ Γ0ij

dxi

ds

dxj

ds= 0 . (7.2)

Substituting (6.8), (5.9), (5.2), (5.12), (6.9), and (6.10) term-by-term, and supposing that the rest-mass of the point-body we areconsidering remains unchanged, we have

m0d2x0

ds2=d

ds

(

m0dx0

ds

)

=d

ds

[mc2

c2−w

(1+

1

c2vj∗uj)]

=

=1

c2−w

(dE

ds+vj

dp j

ds

)

+mc2

(c2−w)2

(1+

1

c2vj∗uj)dwds

+


+m

c2−w∗uj

dvjds=

1

c3

√

1−∗up

∗up

c2

{c2

c2−w

(∗dEdt+vj

dp j

dt

)

+

+mc2

c2 − w

[(1 +

1

c2vn∗un)∗uj

∂vj∂t

+m∗ui ∗uj∂vj∂xi

]

+

+mc4

(c2−w)2

(1+

1

c2vj∗uj)[(1+

1

c2vn∗un) c2

c2−w∂w

∂t+∗ui

∂w

∂xi

]}

.

(7.3)

Because of (3.17), (6.8), and (5.9), we obtain

m0Γ000

dx0

ds

dx0

ds= −

m0

c3c4

(c2 − w)2×

×

[c2

c2 − w∂w

∂t+(1−

w

c2

)vlF

l

](1 + 1

c2vn∗un)2

1−∗up

∗up

c2

=

=−m

c3

√

1−∗up

∗up

c2

c2

c2−w

(1+

1

c2vn∗un)2[ c4

(c2−w)2∂w

∂t+vlF

l

]

.

(7.4)

Next, in the same fashion, because of (3.19), (6.8), (6.18) and(5.9), we obtain

2m0Γ00i

dx0

ds

dxi

ds= 2

m0

c2

[

−c2

c2 − w∂w

∂xi+ vl

(Dli + A

∙li∙+

+1

c2viF

l)] c2

c2 − w

1 + 1c2vn∗un

√

1−∗up

∗up

c2

∗ui

c

√

1−∗uq

∗uq

c2

=

= 2m

c3

√

1−∗up

∗up

c2

c2

c2 − w

(1 +

1

c2vn∗un)×

×

[

−c2

c2 − w∗ui

∂w

∂xi+ vl

(Dli + A

∙li∙

)∗ui +

1

c2vi∗ui(vlF

l)]

.

(7.5)

Finally, because of (3.21), (6.18), (5.9), and also (22.3) of §2.22,

3.7 The time equation of geodesic lines 137

we obtain

m0Γ0ij

dxi

ds

dxj

ds= −

m0

c

c2

c2 − w

{

Ψij −Dij +

+1

c2vl

[(Dlj + A

∙lj∙

)vi +

(Dli + A

∙li∙

)vj +

1

c2vivjF

l]}

×

×∗ui∗uj

c2(1−

∗up∗up

c2

) = −m

c3

√

1−∗up

∗up

c2

c2

c2 − w×

×

[∗ui∗uj

∂vj∂xi

−Dij∗ui ∗uj−

1

c2vj∗uj(Fi∗ui)−∗Δnij

∗ui ∗ujvn +

+2

c2vj∗ujvl

(Dli + A

∙li∙

)∗ui +

1

c4(vj∗uj)2 (

vlFl)]

.

(7.6)

Hence, we have

m0c3(1−

w

c2

)√

1−∗un∗un

c2

(d2x0

ds2+ Γ0μν

dxμ

ds

dxμ

ds

)

=

=∗dE

dt+ vk

∗dpk

dt+m

(1 +

1

c2vn∗un)2( c2

c2 − w

)2 ∂w∂t+

+ m(1 +

1

c2vn∗un) c2

c2 − w∗uj

∂w

∂xj+m

(1 +

1

c2vn∗un)×

×c2

c2 − w∗uj

∂vj∂t

−m(1 +

1

c2vn∗un)2( c2

c2 − w

)2 ∂w∂t+

+ m ∗ui ∗uj∂vj∂xi

−m(1 +

1

c2vn∗un)2 (

vkFk)−

− 2m(1 +

1

c2vn∗un) c2

c2 − w∗ui

∂w

∂xi+ 2m

(1 +

1

c2vn∗un)×

×(Dki + A

∙ki∙

)vk∗ui +mDij

∗ui ∗uj − m∂vj∂xi

∗ui ∗uj +

+ 2m(1 +

1

c2vn∗un)1c2vm

∗um(vkF

k)+1

c2mvn

∗un(Fi∗ui)+

+ mvk∗Δkij

∗ui ∗uj − 2m1

c2vn∗unvk

(Dki + A

∙ki∙

)∗ui −

−1

c4m (vn

∗un)2(vkF

k)=

∗dE

dt+mDij

∗ui ∗uj −mFi∗ui +

+ vk

[ ∗dpk

dt+ ∗Δkij p

i∗uj −mF k + 2m(Dki + A

∙ki∙

)∗ui]

.

(7.7)


Therefore the time equation of geodesics can be written in theform

∗dE

dt+mDij

∗ui∗uj −mFi∗uj +

+ vk

[ ∗dpk

dt+ ∗Δkij p

i ∗uj −mF k + 2m(Dki + A

∙ki∙

)∗ui]

= 0 .

(7.8)

The left side of equation (7.8) is co-invariant, however this for-mula is unchanged under transformation of the time coordinate,because it is a consequence of one of the four geodesic equations,which have world-tensor nature. For this reason we can use themethod to vary the potentials here (see §2.4).

Assuming all the vk are zero, we obtain

∗dE

dt+mDij

∗ui ∗uj −mFj∗uj = 0 . (7.9)

Because the left side of this equation is a chr.inv.-invariant, theequation is true under any choice of the time coordinate (in anyreference frame). So the following equality holds for any choice ofthe time coordinate

vk

[ ∗dpk

dt+ ∗Δkij p


∙ki∙

)∗ui]

= 0 . (7.10)

Assuming term-by-term that

v1 6= 0 , v2 = v3 = 0

v2 6= 0 , v3 = v1 = 0

v3 6= 0 , v1 = v2 = 0

(7.11)

and taking into account that the quantities

∗dpk

dt+ ∗Δkij p


∙ki∙

)∗ui (7.12)

are components of a chr.inv.-vector, we see that the equalities

∗dpk

dt+ ∗Δkij p


∙ki∙

)∗ui = 0 (7.13)

are true for any choice of the time coordinate (in any referenceframe).

3.8 The spatial equations of geodesic lines 139

The four equation we have obtained, namely equation (7.9) andequations (7.13), were deduced from the left side of only one equa-tion — the time equation of geodesics. The other three geodesicequations were not used. At the same time we include the otherthree geodesic equations in the algebra, because only all the fourequations as a whole have world-tensor nature. Therefore we mustconsider equations (7.9) and (7.13) as consequences of all four geo-desic equations, not of only one of them. At the same time thepossibility of obtaining equations (7.9) and (7.13) from only one ofthe geodesic equations demonstrates the utility of the method weused to vary the potentials.

§3.8 The spatial equations of geodesic lines

We are now going to consider the spatial geodesic equations, i. e.equations (7.1) with α = 1, 2, 3

d2xk

ds2+ Γk00

dx0

ds

dx0

ds+ 2Γk0i

dx0

ds

dxi

ds+ Γkij

dxi

ds

dxj

ds= 0 . (8.1)

After the results we have obtained in §3.7, we expect nothingnew from them. For this reason we limit ourselves to their con-sideration in only a formal way.

Employing (3.27), (6.18), (6.20), and (5.9) term-by-term underthe supposition that the rest-mass of the point-body we are con-sidering remains unchanged, we obtain

m0d2xk

ds2+m0Γ

kij

dxi

ds

dxj

ds= m0

(d2xk

ds2+ ∗Δkij

dxi

ds

dxj

ds

)

−

−m0

c2

[(Dkj + A

∙kj∙

)vi +

(Dki + A

∙ki∙

)vj +

1

c2vivjF

k]×

×∗ui∗uj

c2(1−

∗up∗up

c2

) =1

c2

√

1−∗up

∗up

c2

[dpk

dt+ ∗Δkij p

i ∗uj −

−2

c2mvn

∗un(Dki + A

∙ki∙

)∗ui −

1

c4m (vn

∗un)2F k]

.

(8.2)

Because of (3.18), (6.8) and (5.9), we obtain

m0Γk00

dx0

ds

dx0

ds= −

m0

c2

(1−

w

c2

)2F k(

c2

c2 − w

)2×


×

(1 + 1

c2vn∗un)2

1−∗up

∗up

c2

= −m

c2

√

1−∗up

∗up

c2

(1 +

1

c2vn∗un)2F k. (8.3)

In similar fashion, because of (3.20), (6.8), (6.18), and (5.9), weobtain

2m0Γk0i

dx0

ds

dxi

ds= 2

m0

c

(1−

w

c2

)(Dki + A

∙ki∙ +

1

c2viF

k)×

×c2

c2 − w

1 + 1c2vn∗un

c(1−

∗up∗up

c2

)∗ui =

2m

c2

√

1−∗up

∗up

c2

×

×(1 +

1

c2vn∗un)[(

Dki + A

∙ki∙

)∗ui +

1

c2vm

∗umF k]

.

(8.4)

Hence,

m0c2

√

1−∗up∗up

c2

(d2xk

ds2+ Γkμν

dxμ

ds

dxν

ds

)

=

=∗dpk

dt+ ∗Δkij p

i ∗uj −2

c2mvn

∗un(Dki + A

∙ki∙

)∗ui −

−1

c4m (vn

∗un)2F k −m

(1 +

1

c2vn∗un)2F k +

+ 2m(1 +

1

c2vn∗un)(Dki + A

∙ki∙

)∗ui +

+ 2m(1 +

1

c2vn∗un) 1c2vm

∗umF k =

=∗dpk

dt+ ∗Δkij p


∙ki∙

)∗ui.

(8.5)

Therefore the spatial geodesic equations can be written in theform

∗dpk

dt+ ∗Δkij p

i∗uj −mF k + 2m(Dki + A

∙ki∙

)∗ui = 0 , (8.6)

which coincides with formula (7.13).Because of the identity

gμνdxμ

ds

dxμ

ds= 1 , (8.7)

only three of the four geodesic equations are independent. Accord-

3.9 The mechanical sense of the power quantities 141

ingly, only three of the four equations (7.9) and (7.13) are inde-pendent. The identity which links equations (7.9) and (7.13) canbe obtained from the condition (8.7) or, equivalently, from formula(4.3). As a result, we obtain

(m0c√g00

dx0ds

)2− hik

(

m0cdxi

ds

)(

m0cdxk

ds

)

= m0c2 (8.8)

and, taking (5.8), (5.2), and (5.11) into account,

E2

c2− hik p

ipk = m20 c2 (8.9)


E2

c2− pj p

j = m20 c2. (8.10)

Formula (7.13) can be considered as the main equations for thedynamics of a point-mass, and also as the theorem of momentum.Formula (7.9) can be considered as the theorem of energy, whichis a consequence of the main equations (7.13), because of equation(8.9) or equation (8.10).

Let us show how to obtain (7.9) from (7.13), using (8.9) andunder the supposition that m0 remains unchanged. Under this sup-position, equation (8.9) leads to

2

c2E∗dE

dt− pipk

∗dhikdt

− 2hik pi∗dpk

dt= 0 . (8.11)

Because of (6.2) and also (13.11) of §2.13, we have∗dhikdt

= 2Dik + (∗Δij,k +

∗Δkj,i)∗uj . (8.12)

For this reason, we obtain∗dE

dt= mDij

∗ui∗uj +

( ∗dpk

dt+ ∗Δkij p

i ∗uj)∗uk. (8.13)

Finally, considering (8.13) and (7.13) together, we obtain (7.9).

§3.9 The mechanical sense of the power quantities

We introduce a chr.inv.-vector Ωi, defining it by the equality∗

Ωi =1

2εijkA

jk. (9.1)

∗Compare this definition with formula (16.9) of §2.16.


Then (see formula 16.13 of §2.16) we have

Ajk = εijkΩi , (9.2)

and we can write the vector product of Ωi and ∗uj as follows

εijkΩi∗uj = Ajk ∗uj = A∙ki∙

∗ui. (9.3)

Therefore we can write (7.13) in the form

∗dpk

dt+ ∗Δkij p

i∗uj = mF k − 2mεijkΩi∗uj − 2mD

ki∗ui. (9.4)

Equation (9.4) and also equation (7.9), namely

∗dE

dt= mFj

∗uj −mDij∗ui ∗uj , (9.5)

retain their form in all reference frames, and hence, also in theat any given point locally-stationary reference frame. However thechr.inv.-tensor Dik, by its very definition, is zero in the locally-stationary reference frame, so we have, at the given point of thisreference frame

∗dpk

dt+ ∗Δkij p

i∗uj = mF k − 2mεijkΩi∗uj , (9.6)

∗dE

dt= mFj

∗uj . (9.7)

Hence, setting the potentials to zero at the world-point we areconsidering, we have at this point

dpk

dt+Δkij p

iuj = mF k − 2mεijkΩiuj , (9.8)

dE

dt= mFju

j . (9.9)

On the other hand, the equations in an arbitrary referenceframe have the classic form (in curvilinear coordinates)∗

dpk

dt+Δkij p

iuj = φk − 2mεijkψi uj , (9.10)

dE

dt= φju

j , (9.11)

∗As a mater of fact, the relativistic definitions of energy (its varying part)and momentum coincide with the classic definitions only under the approxima-tion uk→ 0.

3.10 The energy-momentum tensor 143

where φk and φj are the contravariant and covariant vectors ofa force (the force includes forces of inertia), ψi is the covariantvector of the angular velocities of the rotation of the given referenceframe. Therefore we can say that F k and Fj play the part of thestrength of gravitational inertial force fields or, equivalently, thepart of the gravitational inertial force, calculated for a unit mass.Thus, Ωi plays the part of the momentary chr.inv.-angular velocityof the absolute rotation of the reference frame at the given point.

§3.10 The energy-momentum tensor

We are going to consider the energy-momentum world-tensor inthe coordinates xσ (σ=0, 1, 2, 3), which set the potentials to zeroat the world-point we are considering. At this world-point, in acontinuous medium, we have

T 00 = ρ , (10.1)

T 0k =1

cJ k, (10.2)

T ij =1

c2U ij , (10.3)

where ρ is the density of the moving substance, Jk is the momen-tum density (or, equivalently, the density of the mass flux), U ij isthe three-dimensional tensor of the kinematic (absolute) stresses,which is the sum of the tensor of the regular (relative) stresses andthe tensor of the density of the momentum flux∗. The world-pointwe are considering is characterized by the conditions

g00 = 1 , g0i = 0 , (10.4)

so at this world-point we have

T00g00

=g0α g0β T

αβ

g00= T 00 (10.5)

and alsoT00g00

= ρ . (10.6)

∗See [8], p. 231, and also [1], p. 70. Further, we will use ρ to denote the “localdensity” of matter. In the homogeneous models, regular density and pressurecoincide with the local density and pressure.


In similar fashion, and taking (10.4) into account, we obtain

T k0√g00

=g0αT

αk

√g00

= T 0k (10.7)

and hence,T k0√g00

=1

cJ k. (10.8)

In accordance with §2.3, the quantities on the left sides of (10.6),(10.8), (10.3) are a chr.inv.-invariant, a chr.inv.-vector, and a chr.inv.-tensor, respectively. Therefore, we can write

T00g00

=T00g00

. (10.9)

T k0√g00

=T k0√g00

, (10.10)

T ij = T ij , (10.11)

so, in general, we haveT00g00

= ρ , (10.12)

T k0√g00

=1

cJ k, (10.13)

T ij =1

c2U ij . (10.14)

Let us find formulas for the components of the covariant, mixed,and contravariant energy-momentum tensors. Because of (10.2),we obtain

T00 =(1−

w

c2

)2ρ . (10.15)

Because of (10.13), we obtain

T k0 =1

c

(1−

w

c2

)J k. (10.16)

SinceT00 = g0αT

α0 = g00T

00 + g0kT

k0 , (10.17)

and taking (10.15) and (10.16) into account, we obtain(1−

w

c2

)2ρ =

(1−

w

c2

)2T 00 −

1

c2

(1−

w

c2

)2vkJ

k, (10.18)

3.10 The energy-momentum tensor 145

T 00 = ρ+1

c2vjJ

j . (10.19)

In similar fashion, because

T k0 = g0αTαk = g00T

0k + g0jTjk, (10.20)


1

c

(1−

w

c2

)J k =

(1−

w

c2

)2T 0k −

vjc

(1−

w

c2

) 1c2U jk, (10.21)

T 0k =1

c

c2

c2 − w

(J k +

1

c2vjU

jk). (10.22)

On the other hand, we have

T k0 = gkαTα0 = gk0T00 + gkjTj0 , (10.23)

so, taking (10.16) and (10.15) into account, we obtain

1

c

(1−

w

c2

)J k = −

1

c

(1−

w

c2

)vkρ− hkjTj0 , (10.24)

T0i = −1

c

(1−

w

c2

)(Ji + ρvi) . (10.25)

We are similarly led to

T ij = giαT jα = gi0Tj0 + g

ikTjk , (10.26)

which, taking (10.14) and (10.16) into account, gives

1

c2U ij = −

1

c2viJ i − hikT jk , (10.27)

Tji = −

1

c2

(viJ

j + Uji

). (10.28)

We now use (10.19), (10.25), and (10.28) for deducing the otherforms of the energy-momentum tensor. Because

T 00 = g0αTα0 = g00T

00 + g0kTk0, (10.29)


ρ+1

c2vjJ

j =(1−

w

c2

)2T 00 −

1

c2vk

(J k +

1

c2vjU

jk), (10.30)


T 00 =

(c2

c2 − w

)2(ρ+

2

c2vjJ

j +1

c4vjvkU

jk). (10.31)

BecauseT0i = g0αT

αi = g00T

0i + g0j T

ji , (10.32)


−1

c

(1−

w

c2

)(Ji + ρvi

)=

=(1−

w

c2

)2T 0i +

vjc

(1−

w

c2

) 1c2

(viJ

j + Uji

),

(10.33)

T 0i = −1

c

c2

c2 − w

[

Ji +(ρ+

1

c2vjJ

j)vi +

1

c2vjU

ji

]

. (10.34)

Finally, because

Tji = gjαTαi = gj0T0i + g

jkTki , (10.35)


−1

c2

(viJ

j + Uji

)=1

c2(Ji + ρvi) v

j − hjk Tki , (10.36)

Tij =1

c2( ρvivj + viJj + vjJi + Uij) . (10.37)

To see the results clearly, we collect formulae (10.15), (10.25),and (10.37)

T00 =(1−

w

c2

)2ρ

T0i = −1

c

(1−

w

c2

)(Ji + ρvi)

Tij =1

c2( ρvivj + viJj + vjJi + Uij)

, (10.38)

formulae (10.19), (10.16), (10.34), and (10.28)

T 00 = ρ+1

c2vjJ

j

T k0 =1

c

(1−

w

c2

)J k

T 0i = −1

c

c2

c2 − w

[

Ji+(ρ+

1

c2vjJ

j)vi+

1

c2vjU

ji

]

Tji = −

1

c2

(viJ

j + Uji

)

, (10.39)

3.11 The time equation of the law of energy 147

and finally, formulae (10.31), (10.22), and (10.14)

T 00 =

(c2

c2 − w

)2(ρ+

2

c2vjJ

j +1

c4vjvkU

jk)

T 0k =1

c

c2

c2 − w

(J k +

1

c2vjU

jk)

T ij =1

c2U ij

. (10.40)

Let us deduce a formula for the chr.inv.-invariant

ρ0 = T = gμνTμν = gμνTμν = T νν . (10.41)

BecauseT νν = T 00 + T

jj , (10.42)

and denotingU = U

jj , (10.43)

we have

T = ρ−1

c2U (10.44)


ρ0 = ρ−1

c2U . (10.45)

§3.11 The time equation of the law of energy

We assume the law of energy to be the divergence of the mixedenergy-momentum tensor equated to zero

∂T νμ∂xν

− Γσμν Tνσ +

∂ ln√−g

∂xσT σμ = 0 . (11.1)

This gives four equations. We consider first the time equation(μ = 0)

∂T ν0∂xν

− Γσ0ν Tνσ +

∂ ln√−g

∂xσT σ0 = 0 . (11.2)


∂T ν0∂xν

=∂T 00∂x0

+∂T

j0

∂xj=


=1

c

∂ρ

∂t+1

c3J j∂vj∂t

+1

c3vj∂J j

∂t−1

c3J j∂w

∂xj+

+1

c

(1−

w

c2

)∂J j

∂xj=1

c

(1−

w

c2

)( ∗∂ρ

∂t+

∗∂J j

∂xj−1

c2FjJ

j

)

.

(11.3)

Besides these, using (3.17–3.20), we obtain

−Γσ0ν Tνσ = −Γ

000T

00 − Γ

k00T

0k − Γ

00iT

i0 − Γ

k0iT

ik =

=1

c3

[c2

c2 − w∂w

∂t+(1−

w

c2

)vlF

l

](ρ+

1

c2vjF

j)−

−1

c2

(1−

w

c2

)2F k

1

c

c2

c2−w

[

Jk+(ρ+

1

c2vjF

j)vk+

1

c2vjU

jk

]

−

−1

c2

[

−c2

c2−w∂w

∂xi+vl

(Dli+A

∙li∙+

1

c2viF

l)] 1c

(1−

w

c2

)J i+

+1

c

(1−

w

c2

)(Dki + A

∙ki∙ +

1

c2viF

k) 1c2(vkJ

i + U ik)=

=1

c3

[c2

c2 − w

(ρ+

1

c2vjJ

j)∂w∂t−(1−

w

c2

)FjJ

j+

+ J i∂w

∂xi+(1−

w

c2

)DikU

ik

]

.

(11.4)

We finally get

∂ ln√−g

∂xσT σ0 =

∂ ln√−g

∂x0T 00 +

∂ ln√−g

∂xjTj0 =

=1

c

[

−1

c2c2

c2 − w∂w

∂t+(1−

w

c2

)D

](ρ+

1

c2vjJ

j)+

+1

c

(

−1

c2c2

c2 − w∂w

∂xj+∂ ln

√h

∂xj

)(1−

w

c2

)J j =

= −

[1

c3c2

c2 − w∂w

∂t

(ρ+

1

c2vjJ

j)+ J j

∂w

∂xj

]

+

+1

c

(1−

w

c2

)(

ρD + J j∗∂ ln

√h

∂xj

)

,

(11.5)

3.12 The spatial equations of the law of energy 149

hence

∂T ν0∂xν

− Γσ0ν Tνσ +

∂ ln√−g

∂xσT σ0 =

=1

c

(1−

w

c2

)( ∗∂ρ

∂t+ ρD + ∗∇jJ

j −2

c2FjJ

j +1

c2DikU

ik

) (11.6)

and (11.2) can be written in the form

∗∂ρ

∂t+ ρD + ∗∇jJ

j −2

c2FjJ

j +1

c2DikU

ik = 0 . (11.7)

§3.12 The spatial equations of the law of energy

We consider next the spatial equations of the law of energy (11.1),i. e. the equations with μ = 1, 2, 3

∂T νi∂xν

− Γσiν Tνσ +

∂ ln√−g

∂xσT σi = 0 . (12.1)

Following §3.11, we obtain

∂T νi∂xν

=∂T 0i∂x0

+∂T

ji

∂xj=−

1

c2c2

(c2−w)2

[

Ji+(ρ+

1

c2vjJ

j)vi+

+1

c2vjU

ji

]∂w

∂t−1

c2c2

c2 − w

[∂Ji∂t

+(ρ+

1

c2vjJ

j)∂vi∂t

+

+ vi

(∂ρ

∂t+1

c2J j∂vj∂t+1

c2vj∂J j

∂t

)

+1

c2Uji

∂vj∂t+1

c2vj∂U

ji

∂t

]

−

−1

c2J j∂vi∂xj

−1

c2vi∂J j

∂xj−1

c2∂U

ji

∂xj= −

1

c2c2

(c2 − w)2×

×

[

Ji +(ρ+

1

c2vjJ

j)vi +

1

c2vjU

ji

]∂w

∂t−1

c2c2

c2 − w×

×

[(ρ+

1

c2vjJ

j)∂vi∂t+1

c2Uji

∂vj∂t

]

−1

c2

∗∂Ji∂t

−1

c2

∗∂Uji

∂xj−

−1

c2vi

( ∗∂ρ

∂t+

∗∂J j

∂xj+1

c2c2

c2 − wJ j

∗∂vj∂t

)

−1

c2J j

∗∂vi∂xj

,

(12.2)


Hence, we get

−Γσiν Tνσ = −Γ

0i0 T

00 − Γ

ki0 T

0k − Γ

0ij T

j0 − Γ

kij T

jk =

= −1

c2

[

−c2

c2 − w∂w

∂xi+ vl

(Dli + A

∙li∙ +

1

c2viF

l)×

×(ρ+

1

c2vjJ

j)+1

c

(1−

w

c2

)(Dki + A

∙ki∙ +

1

c2viF

k)×

×1

c

c2

c2 − w

[

Jk +(ρ+

1

c2vjJ

j)vk +

1

c2vjU

jk

]

+

+1

c

c2

c2 − w

{

Ψij −Dij +1

c2vl

[(Dlj + A

∙lj∙

)vi +

+(Dli + A

∙li∙

)vj +

1

c2vivjF

l]} 1c

(1−

w

c2

)J j +

{∗Δkij −

−1

c2

[(Dkj + A

∙kj∙

)vi +

(Dki + A

∙ki∙

)vj +

1

c2vivjF

k]}

×

×1

c2

(vkJ

j + Ujk

)=1

c2c2

c2 − w

(ρ+

1

c2vjJ

j) ∂w∂xi

+

+1

cJ j∂vj∂xi

+1

c2∗Δkij U

jk −

1

c4viDjkU

jk.

(12.3)

Finally,

∂ ln√−g

∂xσT σi =

∂ ln√−g

∂x0T 0i +

∂ ln√−g

∂xjTji =

=1

c

[

−1

c2c2

c2 − w∂w

∂t+(1−

w

c2

)D

]1

c

c2

c2 − w×

×

[

Ji +(ρ+

1

c2vjJ

j)vi +

1

c2vjU

ji

]

−

−

(

−1

c2c2

c2 − w∂w

∂xj+∂ ln

√h

∂xj

)1

c2

(viJ

j + Uji

)=

=1

c2c2

(c2−w)2

[

Ji +(ρ+

1

c2vjJ

j)vi +

1

c2vjU

ji

]∂w

∂t−

−1

c2DJi −

1

c2Uji

∗∂ ln√h

∂xj+1

c4c2

c2 − wUji

∂w

∂xj−

−1

c2vi

(

ρD + J j∗∂ ln

√h

∂xj−1

c2c2

c2 − wJ j∂w

∂xj

)

.

(12.4)

3.12 The spatial equations of the law of energy 151

Hence, we have

∂T νi∂xν

− Γσiν Tνσ +

∂ ln√−g

∂xσT σi =

1

c2

[(ρ+

1

c2vj J

j)Fi +

+1

c2FjU

ji −

∗∂Ji∂t

−DJi −∗∇j U

ji +

(∂vj∂xi

−∂vi∂xj

)

J j]

−

−1

c2vi

[ ∗∂ρ∂t

+ ρD + ∗∇jJj −

1

c2FjJ

j +1

c2Djk U

jk

]

=

= −1

c2

{[ ∗∂Ji∂t

+DJi+∗∇j U

ji −2AijJ

j−Fj(ρh

ji+U

ji

)]

+

+ vi

[ ∗∂ρ∂t

+ ρD + ∗∇j Jj −

2

c2FjJ

j +1

c2Djk U

jk

]}

(12.5)

and (12.1) can be written in the form( ∗∂Ji∂t

+DJi +∗∇j U

ji − 2AijJ

j − ρFi −1

c2FjU

ji

)

+

+ vi

( ∗∂ρ

∂t+ ρD + ∗∇jJ

j −2

c2FjJ

j +1

c2DjkU

jk

)

= 0 .

(12.6)

With the remaining formula (11.7) unused, we apply the methodto vary the potentials.

Equation (12.6) retains its form in transformations of the timecoordinate, although its left side is a co-vector, not a chr.inv.-vector.For this reason we can set all values of vi to zero. Then we have

∗∂Ji∂t

+DJi +∗∇j U

ji − 2AijJ

j − ρFi −1

c2FjU

ji = 0 , (12.7)

and this sub-vector equation is valid for any choice of the timecoordinate, because its left side is a chr.inv.-vector. Thus, for anychoice of the time coordinate, we have

vi

( ∗∂ρ

∂t+ ρD + ∗∇jJ

j −2

c2FjJ

j +1

c2Djk U

jk

)

= 0 (12.8)

and hence, we obtain the equation

∗∂ρ

∂t+ ρD + ∗∇jJ

j −2

c2FjJ

j +1

c2Djk U

jk = 0 , (12.9)

which coincides with formula (11.7).


So, just as for the geodesic equations, the method of varying thepotentials has found four equations from only a subset of them (seethe explanation of this fact in §3.7).

Because

hik∗∂Ji∂t

=∗∂J k

∂t−Ji

∗∂hik

∂t=∗∂J k

∂t+2DikJi=

∗∂J k

∂t+2Dk

iJi, (12.10)

we can write

∗∂J k

∂t+DJ k+2

(Dkj +A

∙kj∙

)J j−ρF k−

1

c2FjU

jk+∗∇jUjk=0 (12.11)

instead of equation (12.7).

§3.13 A space element. Its energy and the momentum

Let us take a fixed elementary parallelepiped

Π123abc =

∣∣∣∣∣∣∣

δax1 δax

2 δax3

δbx1 δbx

2 δbx3

δcx1 δcx

2 δcx3

∣∣∣∣∣∣∣

δaxi = constia

δbxi = constib

δcxi = constic

(13.1)

in the space we are considering. We can write its volume V , inaccordance with formula (12.7) of §2.12, as follows

V =√h∣∣Π123abc

∣∣ . (13.2)

The energy E and the momentum pk inside the volume, evi-dently take the form

E = Vρc2, (13.3)

pk = VJ k. (13.4)

Because this elementary volume is fixed in the space, we canwrite ( ∗dE

dt

)

fix

=∗∂E

∂t, (13.5)

( ∗dpk

dt

)

fix

=∗∂pk

∂t. (13.6)

On the other hand, because of formula (12.9) of §2.12, we have

∗∂E

∂t=

( ∗∂ρ

∂t+ ρD

)

c2V , (13.7)

3.13 A space element. Its energy and the momentum 153

∗∂pk

∂t=

( ∗∂J k

∂t+DJ k

)

V . (13.8)

Hence, ( ∗dE

dt

)

fix

=

( ∗∂ρ

∂t+ ρD

)

c2V , (13.9)

( ∗dpk

dt

)

fix

=

( ∗∂J k

∂t+DJ k

)

V . (13.10)

The volume V is a chr.inv.-invariant, which retains its value inthe parallel transfer of the chr.inv.-vectors δaxi, δbxi, δcxi. Natur-ally, because the chr.inv.-tensor of the 3rd rank Πijkabc has the sameproperties as εijk, we have

V =1

6

∣∣∣εijkΠ

ijkabc

∣∣∣ . (13.11)

Because of formula (17.4) of §2.17, we have

∗∇p(εijkΠ

ijkabc

)= εijk

∗∇pΠijkabc . (13.12)

The rule for differentiating determinants holds for chr.inv.-differentiation as well as in covariant differentiation. Hence, theequalities

∗∇p(δax

i)= 0 , ∗∇p

(δbx

i)= 0 , ∗∇p

(δcx

i)= 0 (13.13)

lead to the equality∗∇pΠ

ijkabc = 0 , (13.14)

so we have∗∇pV = 0 . (13.15)

Because of (13.3), (13.4), (13.9), (13.10), and (13.15), and de-noting

m = Vρ (13.16)

for the moving mass throughout this volume, we can, instead of(12.11) and (12.9) write respectively,

(∗dpk

dt

)

fix

−mF k + 2(Dki + A

∙ki∙

)pi =

=1

c2

[Fi(V U ik

)− ∗∇i

(V U ik

)c2],

(13.17)


( ∗dE

dt

)

fix

+Dij(V U ij

)− Fj p

j = Fj pj − ∗∇j p

jc2, (13.18)

where p jc2 is the energy flux.The equations we have obtained here are related to a fixed

volume element, which is the frame of a flowing continuous matter.This is a very important difference to the equations (7.13) and (7.9)we obtained for a free mass-particle, which moves with respect tothe given space.

§3.14 Einstein’s covariant tensor. Its time component

We now consider the Einstein covariant tensor

Gμν = −∂Γαμν∂xα

+ ΓβμαΓανβ +

∂2 ln√−g

∂xμ∂xν− Γαμν

∂ ln√−g

∂xα. (14.1)

We first consider its time component, i. e. the component withthe indices μ, ν =0

G00 = −∂Γα00∂x0

+ Γβ0αΓ

α0β +

∂2 ln√−g

∂x0∂x0− Γα00

∂ ln√−g

∂xα. (14.2)


−∂Γα00∂xα

=−∂Γ000∂x0

−∂Γk00∂xk

=1

c4

[c2

(c2−w)2

(∂w

∂t

)2+

c2

c2−w∂2w

∂t2−

−1

c2vlF

l ∂w

∂t+(1−

w

c2

)F l∂vl∂t

+(1−

w

c2

)vl∂F l

∂t

]

+

+1

c2

(1−

w

c2

)[

−2

c2F k

∂w

∂xk+(1−

w

c2

)∂F k

∂xk

]

=

=1

c4

[c2

(c2 − w)2

(∂w

∂t

)2+

c2

c2 − w∂2w

∂t2−1

c2vlF

l ∂w

∂t

]

−

−1

c4

(1−

w

c2

)F k

∂w

∂xk−1

c2

(1−

w

c2

)2( 1c2FkF

k +∗∂F k

∂xk

)

.

(14.3)

Because of (3.17–3.20), we obtain

Γβ0αΓ

α0β = Γ

000Γ

000 + 2Γ

k00Γ

00k + Γ

k0iΓ

i0k =

=1

c6

[c4

(c2−w)2

(∂w

∂t

)2+ 2vlF

l ∂w

∂t+(1−

w

c2

)2(vlF

l)2]

−

3.14 Einstein’s covariant tensor. Its time component 155

−2

c4

(1−

w

c2

)[

−F k∂w

∂xk+(1−

w

c2

)(Dlk + A

∙lk∙

)vlF

k +

+1

c2

(1−

w

c2

)(vlF

l)2]

+1

c2

(1−

w

c2

)2×

×

[

DkiD

ik + A

∙ki∙A

∙ik∙ +

2

c2(Dki +A

∙ki∙

)vkF

i +1

c4(vkF

k)2]

=

=1

c4

[c2

(c2 − w)2

(∂w

∂t

)2+2

c2vlF

l ∂w

∂t

]

+

+2

c4

(1−

w

c2

)F k

∂w

∂xk+1

c2

(1−

w

c2

)2(DkiD

ik + A

∙ki∙A

∙ik∙

).

(14.4)

Hence, because of (1.7), we obtain

∂2 ln√−g

∂x0∂x0=1

c2∂

∂t

[

−1

c2c2

c2 − w∂w

∂t+(1−

w

c2

)D

]

=

=−1

c4

[c2

(c2−w)2

(∂w

∂t

)2+

c2

c2−w∂2w

∂t2+∂w

∂t

]

+1

c2

(1−

w

c2

)2 ∗∂D∂t

,

(14.5)

and because of (3.17), (3.18), and (1.7), we obtain

−Γα00∂ ln

√−g

∂xα= −Γ000

∂ ln√−g

∂x0− Γk00

∂ ln√−g

∂xk=

=1

c4

[c2

c2−w∂w

∂t+(1−w

c2

)vlF

l

][

−1

c2c2

c2−w∂w

∂t+(1−w

c2

)D

]

+

+1

c2

(1−

w

c2

)2F k(

−1

c2c2

c2 − w∂w

∂xk+∂ ln

√h

∂xk

)

=

= −1

c4

[c2

(c2 − w)2

(∂w

∂t

)2+

(1

c2vlF

l −D

)∂w

∂t

]

−

−1

c4

(1−

w

c2

)F k

∂w

∂xk+1

c2

(1−

w

c2

)2F k

∗∂ ln√h

∂xk.

(14.6)

We finally obtain

G00=1

c2

(1−

w

c2

)2×

×

( ∗∂D

∂t+Dl

jDjl +A

∙lj∙A

∙jl∙+

∗∇jFj−

1

c2FjF

j

)

.

(14.7)


§3.15 Einstein’s covariant tensor. The mixed components

We now consider mixed (space-time) components of the Einsteintensor (14.1), i. e. its components with the indices μ=0, ν=1, 2, 3

G0i = −∂Γα0i∂xα

+ Γβ0αΓ

αiβ +

∂2 ln√−g

∂x0∂xi− Γα0i

∂ ln√−g

∂xα. (15.1)


−∂Γα0i∂xα

= −∂Γ00i∂x0

−∂Γk0i∂xk

= −1

c3

[

−c2

(c2 − w)2∂w

∂t

∂w

∂xi−

−c2

c2 − w∂2w

∂t ∂xi+(Dli + A

∙li∙ +

1

c2viF

l)∂vl∂t

+

+ vl∂

∂t

(Dli + A

∙li∙ +

1

c2viF

l)]

−1

c

[

−1

c2

(Dki + A

∙ki∙ +

+1

c2viF

k) ∂w∂xk

+(1−

w

c2

) ∂

∂xk

(Dki +A

∙ki∙ +

1

c2viF

k)]

=

=1

c3c2

c2−w

(c2

c2−w∂w

∂t

∂w

∂xi+

∂2w

∂t ∂xi

)

+1

c3

(1−

w

c2

)Fk ×

×(Dki + A

∙ki∙ +

1

c2viF

k)−1

c

(1−

w

c2

) ∗∂

∂xk

(Dki + A

∙ki∙ +

+1

c2viF

k)=1

c3c2

c2 − w

(c2

c2 − w∂w

∂t

∂w

∂xi+

∂2w

∂t ∂xi

)

−

−1

c

(1−

w

c2

)[ ∗∂

∂xk(Dki + A

∙ki∙

)−1

c2(Dki + A

∙ki∙

)Fk +

+1

c2F k

∂vi∂xk

+1

c4c2

c2 − wvkF

k ∂vi∂t

]

−

−1

c3vi

(1−

w

c2

)( ∗∂F k

∂xk−1

c2FkF

k

)

.

(15.2)

Because of (3.17–3.22), we obtain

Γβ0αΓ

αiβ = Γ

000Γ

0i0 + Γ

k00Γ

0ik + Γ

00kΓ

ki0 + Γ

j0kΓ

kij =

−1

c5

[c2

c2 − w∂w

∂t+(1−

w

c2

)vkF

k

][

−c2

c2 − w∂w

∂xi+

+(Dli + A

∙li∙ +

1

c2viF

l)vl

]

+1

c3

(1−

w

c2

)F k{

Ψik−Dik +

3.15 Einstein’s covariant tensor. The mixed components 157

+1

c2vl

[(Dlk + A

∙lk∙

)vi +

(Dli + A

∙li∙

)vk +

1

c2vivkF

l]}

+

+1

c3

[

−c2

c2 − w∂w

∂xk+(Dlk + A

∙lk∙ +

1

c2vkF

l)vl

]

×

×(Dki + A

∙ki∙ +

1

c2viF

k)(1−

w

c2

)+1

c

(1−

w

c2

)×

×(Djk + A

∙jk∙ +

1

c2vkF

j){

∗Δkij −1

c2

[(Dkj + A

∙kj∙

)vi+

+(Dki + A

∙ki∙

)vj +

1

c2vivjF

k]}

= −1

c31

c2 − w×

×

[

−c2

c2 − w∂w

∂xi+(Dli + A

∙li∙ +

1

c2viF

l)vl

]∂w

∂t+

+1

c5vnF

n ∂w

∂xi−1

c5

(1−

w

c2

)vnF

n(Dli + A

∙li∙

)vl −

−1

c7

(1−

w

c2

)(vkF

k)2vi −

1

c3

(1−

w

c2

)[

DikFk −

−1

2

(∂vk∂xi

+∂vi∂xk

)

F k +1

2c2(Fkvi + Fivk)F

k +

+ ∗Δlik vlFk −

1

c2vkF

k(Dli + A

∙li∙

)vl

]

+1

c5vi

(1−

w

c2

)×

×

[(Dlk+A

∙lk∙

)vlF

k+1

c2(vkF

k)2]

−1

c3(Dki +A

∙ki∙

) ∂w∂xk

+

+1

c3

(1−

w

c2

)(Dki + A

∙ki∙

)(Dlk + A

∙lk∙

)vl +

1

c5

(1−

w

c2

)×

×(Dki +A

∙ki∙

)vkvlF

l +1

c5vi

(1−

w

c2

)[

−c2

c2−wF k

∂w

∂xk+

+(Dlk + A

∙lk∙

)vlF

k +1

c2(vkF

k)2]

+1

c

(1−

w

c2

)×

×(Djk+A

∙jk∙

)∗Δkij+

1

c3

(1−

w

c2

)∗Δkij vkF

j−1

c3

(1−

w

c2

)×

×(Dki + A

∙ki∙

)(Djk + A

∙jk∙

)vj −

1

c5

(1−

w

c2

)×

×(Dki + A

∙ki∙

)vkvjF

j −1

c3vi

(1−

w

c2

)[

DkjD

jk + A

∙kj∙A

∙jk∙ +


+2

c2(Dkj + A

∙kj∙

)vkF

j +1

c4(vkF

k)2]

= −1

c31

c2 − w×

×

[

−c2

c2 − w∂w

∂xi+ vl

(Dli + A

∙li∙ +

1

c2viF

l)] ∂w

∂t−

−1

c

(1−

w

c2

)[

−∗Δkij(Djk + A

∙jk∙

)−

1

2c2

(∂vk∂xi

+∂vi∂xk

)

F k +

+1

2c4(vkF

k)Fi −

1

c4c2

c2 − wvkF

k ∂w

∂xi+1

c2c2

c2 − w×

×(Dki + A

∙ki∙

) ∂w∂xk

+1

c2DikF

k

]

−1

c3vi

(1−

w

c2

)×

×

(1

2c2FkF

k +1

c2c2

c2 − wF k

∂w

∂xk+Dk

jDjk + A

∙kj∙A

∙jk∙

)

.

(15.3)


∂2 ln√−g

∂x0∂xi=1

c

∂

∂xi

[

−1

c2c2

c2 − w∂w

∂t+(1−

w

c2

)D

]

=

= −1

c3c2

(c2 − w)2∂w

∂xi∂w

∂t−1

c3c2

c2 − w∂2w

∂xi∂t−

−1

c3D∂w

∂xi+(1−

w

c2

)∂D∂xi

.

(15.4)


−Γα0i∂ ln

√−g

∂xα= −Γ00i

∂ ln√−g

∂x0− Γk0i

∂ ln√−g

∂xk=

= −1

c3

[

−c2

c2 − w∂w

∂xi+ vl

(Dli + A

∙li∙ +

1

c2viF

l)]

×

×

[

−1

c2c2

c2 − w∂w

∂t+(1−

w

c2

)D

]

−1

c

(1−

w

c2

)×

×(Dki + A

∙ki∙ +

1

c2viF

k)(

−1

c2c2

c2−w∂w

∂xk+∂ ln

√h

∂xk

)

=

=1

c31

c2−w

[

−c2

c2−w∂w

∂xi+vl

(Dli+A

∙li∙+

1

c2viF

l)]∂w∂t+

+1

c3D∂w

∂xi−1

c

(1−

w

c2

)[(Dki + A

∙ki∙

) ∂ ln√h

∂xk+

3.16 Einstein’s covariant tensor. The spatial components 159

+1

c2vkD

(Dki + A

∙ki∙

)−1

c2c2

c2 − w

(Dki + A

∙ki∙

) ∂w∂xk

]

−

−1

c3vi

(1−

w

c2

)(

F k∂ ln

√h

∂xk+1

c2vkDF

k −

−1

c2c2

c2 − wF k

∂w

∂xk

)

=1

c31

c2 − w

[

−c2

c2 − w∂w

∂xi+

+ vl

(Dli + A

∙li∙ +

1

c2viF

l)]∂w∂t+1

c3D∂w

∂xi−1

c

(1−

w

c2

)×

×

[(Dki + A

∙ki∙

) ∗∂ ln√h

∂xk−1

c2c2

c2 − w

(Dki + A

∙ki∙

) ∂w∂xk

]

−

−1

c3vi

(1−

w

c2

)(

F k∗∂ ln

√h

∂xk−1

c2c2

c2 − wF k

∂w

∂xk

)

.

(15.5)

This gives

G0i = −1

c

(1−

w

c2

)[∗∇k

(Dki + A

∙ki∙

)−∂D

∂xi−1

c2A∙ki∙ Fk−

−1

2c2

(∂vk∂xi

−∂vi∂xk

)

F k−1

2c4Fi vkF

k

]

−1

c3vi

(1−

w

c2

)×

×

(∗∇k F

k−1

2c2FkF

k +DkjD

jk + A

∙kj∙A

∙jk∙

)

.

(15.6)

and so, in the final form, we have

G0i =1

c

(1−

w

c2

)[∗∇j(hjiD −D

ji

)− ∗∇jA

∙ji∙ +

2

c2AijF

j−

−1

c2vi

( ∗∂D

∂t+Dl

jDjl + A

∙lj∙A

∙jl∙ +

∗∇j Fj −

1

c2FjF

j

)]

.

(15.7)

§3.16 Einstein’s covariant tensor. The spatial components

We finally consider spatial components of the Einstein tensor (14.1),i. e. its components with μ, ν =1,2,3

Gij = −∂Γαij∂xα

+ ΓβiαΓ

αjβ +

∂2 ln√−g

∂xi∂xj− Γαij

∂ ln√−g

∂xα. (16.1)

Because it will entail more difficult algebra than that used for G00and G0i, we will summarize all the necessary terms step-by-step.


First, we have

−∂Γαij∂xα

= −∂Γ0ij∂x0

−∂Γkij∂xk

. (16.2)

Because of (3.21) and also (22.3) of §2.22, we obtain

−∂Γ0ij∂x0

=1

c2c2

(c2 − w)2

{

Ψij −Dij +

+1

c2vl

[(Dlj + A

∙lj∙

)vi + (D

li + A

∙li∙

)vj +

1

c2vivjF

l]}∂w

∂t+

+1

c2c2

c2 − w

(∂Ψij∂t

−∂Dij

∂t

)

+1

c4c2vlc2 − w

×

×

[(Dlj + A

∙lj∙

)∂vi∂t

+(Dli + A

∙li∙

)∂vj∂t

]

+1

c4c2vic2 − w

×

×

[(Dlj + A

∙lj∙

)∂vl∂t

+ vl∂

∂t

(Dlj + A

∙lj∙

)+1

c2vlF

l ∂vj∂t

]

+

+1

c4c2vjc2 − w

[(Dli + A

∙li∙

)∂vl∂t

+ vl∂

∂t

(Dli + A

∙li∙

)+

+1

c2vlF

l ∂vi∂t

]

+1

c6c2vivjc2 − w

(

F l∂vl∂t

+ vl∂F l

∂t

)

=

= (α1)ij + (β1)ij + (γ1)ij + (δ1)ij + (ε1)ij ,

(16.3)

where the quantities (α1)ij , (β1)ij , (γ1)ij , (δ1)ij , (ε1)ij , are given by

(α1)ij =1

c2c2

(c2 − w)2

{

Ψij −Dij +1

c2vl

[(Dlj + A

∙lj∙

)vi+

+(Dli + A

∙li∙

)vj +

1

c2vivjF

l]} ∂w

∂t,

(16.4)

(β1)ij =1

c2

{

−∗∂Dij

∂t+

∗∂

∂t

[1

2

(∂vj∂xi

+∂vi∂xj

)

−

−1

2c2(Fjvi + Fivj)

]

−∗Δlijc2

c2 − w∂vl∂t

+ vl

[

−∗∂ ∗Δlij∂t

+

+1

c2c2

c2 − w

(Dlj + A

∙lj∙

)∂vi∂t

+1

c2c2

c2 − w

(Dli + A

∙li∙

)∂vj∂t

]}

,

(16.5)


(γ1)ij=1

c4c2vic2−w

[(Dlj+A

∙lj∙

)∂vl∂t+ vl

∂

∂t

(Dlj+A

∙lj∙

)+1

c2vlF

l ∂vj∂t

]

, (16.6)

(δ1)ij=1

c4c2vjc2−w

[(Dli+A

∙li∙

)∂vl∂t+ vl

∂

∂t

(Dli+A

∙li∙

)+1

c2vlF

l ∂vi∂t

]

, (16.7)

(ε1)ij =1

c6c2vivjc2 − w

(

F l∂vl∂t

+ vl∂F l

∂t

)

. (16.8)


−∂Γkij∂xk

= −∗∂ ∗Δkij∂xk

+1

c2vl

∗∂ ∗Δlij∂t

+1

c2

[(Dkj + A

∙kj∙

) ∂vi∂xk

+

+(Dki +A

∙ki∙

) ∂vj∂xk

]

+1

c2vi

[∂

∂xk(Dkj +A

∙kj∙

)+1

c2F k

∂vj∂xk

]

+

+1

c2vj

[∂

∂xk(Dki + A

∙ki∙

)+1

c2F k

∂vi∂xk

]

+1

c4vivj

∂F k

∂xk=

= (β2)ij + (γ2)ij + (δ2)ij + (ε2)ij ,

(16.9)

where we denote

(β2)ij = −∗∂ ∗Δkij∂xk

+1

c2vl

∗∂ ∗Δlij∂t

+

+1

c2

[(Dkj + A

∙kj∙

) ∂vi∂xk

+(Dki + A

∙ki∙

) ∂vj∂xk

]

,

(16.10)

(γ2)ij =1

c2vi

[∂

∂xk(Dkj + A

∙kj∙

)+1

c2F k

∂vj∂xk

]

, (16.11)

(δ2)ij =1

c2vj

[∂

∂xk(Dki + A

∙ki∙

)+1

c2F k

∂vi∂xk

]

, (16.12)

(ε2)ij =1

c4vivj

∂F k

∂xk. (16.13)

The second term of (16.1) is

ΓβiαΓ

αjβ = Γ

0i0Γ

0j0 + Γ

ki0Γ

0jk + Γ

0ikΓ

kj0 + Γ

likΓ

kjl . (16.14)


Γ0i0Γ0j0 =

1

c4

[c2

(c2−w)2∂w

∂xi∂w

∂xj−vl

(Dli+A

∙li∙

) c2

c2−w∂w

∂xj−


−1

c2vivlF

l c2

c2 − w∂w

∂xj− vl

(Dlj + A

∙lj∙

) c2

c2 − w∂w

∂xi−

−1

c2vjvlF

l c2

c2 − w∂w

∂xi+ vk

(Dki + A

∙ki∙

)vl(Dlj + A

∙lj∙

)+

+1

c2vivlF

lvk(Dkj + A

∙kj∙

)+1

c2vjvlF

lvk(Dki + A

∙ki∙

)+

+1

c4vivj

(vlF

l)2]

= (β3)ij + (γ3)ij + (δ3)ij + (ε3)ij ,

(16.15)

where we denote

(β3)ij =1

c4

[c4

(c2−w)2∂w

∂xi∂w

∂xj− vl

(Dli+A

∙li∙

) c2

c2−w∂w

∂xj−

− vl(Dlj +A

∙lj∙

) c2

c2−w∂w

∂xi+ vk

(Dki +A

∙ki∙

)vl(Dlj +A

∙lj∙

)]

,

(16.16)

(γ3)ij =1

c6vivlF

l

[

−c2

c2 − w∂w

∂xj+ vk

(Dkj + A

∙kj∙

)]

, (16.17)

(δ3)ij =1

c6vjvlF

l

[

−c2

c2 − w∂w

∂xi+ vk

(Dki + A

∙ki∙

)]

, (16.18)

(ε3)ij =1

c8vivj

(vlF

l)2. (16.19)

Because of (3.20), (3.21), and also (22.3) of §2.22, we obtain

Γki0Γ0jk + Γ

0ikΓ

kj0 = −

1

c2

[(Dki + A

∙ki∙

)(Ψjk −Djk

)+

+1

c2vi (Ψjk−Djk)F

k +1

c2vjvl

(Dki + A

∙ki∙

)(Dlk + A

∙lk∙

)+

+1

c4vjvlF

lvk(Dki +A

∙ki∙

)+1

c2vkvl

(Dki +A

∙ki∙

)(Dlj+A

∙lj∙

)+

+1

c4vivjvl

(Dlk + A

∙lk∙

)F k +

1

c6vivj

(vlF

l)2+

+1

c4vivlF

lvk(Dkj + A

∙kj∙

)+(Dkj + A

∙kj∙

)(Ψik −Dik)+

+1

c2vj (Ψik−Dik)F

k +1

c2vivl

(Dkj + A

∙kj∙

)(Dlk + A

∙lk∙

)+

+1

c4vivlF

lvk(Dkj +A

∙kj∙

)+1

c2vkvl

(Dkj +A

∙kj∙

)(Dli+A

∙li∙

)+


+1

c4vivjvl

(Dlk + A

∙lk∙

)F k +

1

c6vivj

(vlF

l)2+

+1

c4vjvlF

lvk(Dki +A

∙ki∙

)]

=(β4)ij+(γ4)ij+(δ4)ij+(ε4)ij ,(16.20)

where we denote

(β4)ij =1

c2

{

2DkiDjk + A

∙ki∙Djk + A

∙kj∙Dik−

−(Dki + A

∙ki∙

)[1

2

(∂vk∂xj

+∂vj∂xk

)

−1

2c2(Fjvk + Fkvj)

]

−

−(Dkj + A

∙kj∙

)[1

2

(∂vk∂xi

+∂vi∂xk

)

−1

2c2(Fivk + Fkvi)

]

+

+ ∗Δljk vl(Dki + A

∙ki∙

)+ ∗Δlik vl

(Dkj + A

∙kj∙

)−

−2

c2vkvl

(Dki + A

∙ki∙

)(Dlj + A

∙lj∙

)}

,

(16.21)

(γ4)ij =1

c4vi

{

DjkFk−

−

[1

2

(∂vk∂xj

+∂vj∂xk

)

−1

2c2(Fjvk+Fkvj)

]

F k+∗Δljk vlFk −

−2

c2vkF

kvl(Dlj + A

∙lj∙

)− vl

(Dkj + A

∙kj∙

)(Dlk + A

∙lk∙

)}

,

(16.22)

(δ4)ij =1

c4vj

{

DikFk −

−

[1

2

(∂vk∂xi

+∂vi∂xk

)

−1

2c2(Fivk + Fkvi)

]

F k+∗Δlik vlFk −

−2

c2vkF

kvl(Dli + A

∙li∙

)− vl

(Dki + A

∙ki∙

)(Dlk + A

∙lk∙

)}

,

(16.23)

(ε4)ij =1

c6vivj

[

−2vl(Dlk + A

∙lk∙

)F k −

2

c2(vlF

l)2]

. (16.24)


ΓlikΓkjl =

∗Δlik∗Δkjl−

−1

c2∗Δlik

[(Dkl + A

∙kl∙

)vj +

(Dkj + A

∙kj∙

)vl +

1

c2vjvlF

k

]

−


−1

c2∗Δkjl

[(Dlk + A

∙lk∙

)vi +

(Dli + A

∙li∙

)vk +

1

c2vivkF

l

]

+

+1

c4

[(DlkD

kl +A

∙lk∙A

∙kl∙

)vivj + vivl

(Dkj +A

∙kj∙

)(Dlk+A

∙lk∙

)+

+1

c2vivjvl

(Dlk + A

∙lk∙

)F k + vjvk

(Dli + A

∙li∙

)(Dkl + A

∙kl∙

)+

+ vlvk(Dli + A

∙li∙

)(Dkj + A

∙kj∙

)+1

c2vjvkF

kvl(Dli + A

∙li∙

)+

+1

c2vivjvk

(Dkl + A

∙kl∙

)F l +

1

c2vivlF

lvk(Dkj + A

∙kj∙

)+

+1

c4vivj

(vlF

l)2]

= (β5)ij + (γ5)ij + (δ5)ij + (ε5)ij ,

(16.25)

where we denote

(β5)ij =∗Δlik

∗Δkjl −1

c2∗Δlik vl

(Dkj + A

∙kj∙

)−

−1

c2∗Δkjl vk

(Dli + A

∙li∙

)+1

c4vlvk

(Dli + A

∙li∙

)(Dkj + A

∙kj∙

),

(16.26)

(γ5)ij =1

c5vi

[

−∗Δkjl(Dlk + A

∙lk∙

)−1

c2∗Δkjl vkF

l+

+1

c2vl(Dkj + A

∙kj∙

)(Dlk + A

∙lk∙

)+1

c4vlF

lvk(Dkj + A

∙kj∙

)]

,

(16.27)

(δ5)ij =1

c2vj

[

−∗Δlik(Dkl + A

∙kl∙

)−1

c2∗Δlik vlF

k+

+1

c2vk(Dli + A

∙li∙

)(Dkl + A

∙kl∙

)+1

c4vkF

kvl(Dli + A

∙li∙

)]

,

(16.28)

(ε5)ij =1

c4vivj×

×

[

DlkD

kl + A

∙lk∙A

∙kl∙ +

2

c2vl(Dlk + A

∙lk∙

)F k +

1

c4(vlF

l)2]

.(16.29)


∂2 ln√−g

∂xi∂xj=

∂

∂xi

(

−1

c2c2

c2−w∂w

∂xj+∂ ln

√h

∂xj

)

=

= −1

c2c2

(c2−w)2∂w

∂xi∂w

∂xj−1

c2c2

c2−w∂2w

∂xi∂xj+∂2 ln

√h

∂xi∂xj.

(16.30)


On the other hand, the last term is

∗∂2 ln√h

∂xi∂xj=

∗∂

∂xi

(∗∂ ln√h

∂xj

)

=∗∂

∂xi

(∂ ln

√h

∂xj+1

cvjD

)

=

=∂2 ln

√h

∂xi∂xj+

1

c2 − wvj

∂

∂xi

[(1−

w

c2

)D]+

+1

c2

(∂vi∂xj

+vj

c2 − w∂vi∂t

)

D +1

c2vi∗∂D

∂xj=∂2 ln

√h

∂xi∂xj−

−1

c4vj

c2

c2 − wD∂w

∂xi+1

c2vj∂D

∂xi+1

c2D∂vi∂xj

+

+1

c4vj

c2

c2 − wD∂vi∂t

+1

c2vi∗∂D

∂xj=∂2 ln

√h

∂xi∂xj+

+1

c2D

(∂vi∂xj

−1

c2Fivj

)

+1

c2

(

vi∗∂D

∂xj+vj

∗∂D

∂xi

)

−1

c4vivj

∗∂D

∂t. (16.31)


∂2 ln√−g

∂xi∂xj= −

1

c2c2

(c2−w)2∂w

∂xi∂w

∂xj−1

c2c2

c2−w∂2w

∂xi∂xj+

+∗∂2 ln

√h

∂xi∂xj−1

c2D

(∂vi∂xj

−1

c2Fivj

)

−1

c2

(

vi∗∂D

∂xj+vj

∗∂D

∂xi

)

+

+1

c4vivj

∗∂D

∂t= (β6)ij + (γ6)ij + (δ6)ij + (ε6)ij ,

(16.32)

where we denote

(β6)ij =∗∂2 ln

√h

∂xj∂xi−1

c2

[c2

(c2 − w)2∂w

∂xi∂w

∂xj+

+c2

c2 − w∂2w

∂xi∂xj+D

(∂vi∂xj

−1

c2Fivj

)]

,

(16.33)

(γ6)ij = −1

c2vi∗∂D

∂xj, (16.34)

(δ6)ij = −1

c2vj∗∂D

∂xi, (16.35)

(ε6)ij =1

c4vivj

∗∂D

∂t. (16.36)


Finally, we have the term

−Γαij∂ ln

√−g

∂xα= −Γ0ij

∂ ln√−g

∂x0− Γkij

∂ ln√−g

∂xk. (16.37)

Because of (3.21), (1.7), and also (22.3) of §3.22, we obtain

−Γ0ij∂ ln

√−g

∂x0=1

c2c2

c2−w

{

Ψij−Dij+1

c2vl

[(Dlj+A

∙lj∙

)vi+

+(Dli+A

∙li∙

)vj+

1

c2vivjF

l]}[

−1

c2c2

c2−w∂w

∂t+(1−

w

c2

)D

]

=

= (α7)ij + (β7)ij + (γ7)ij + (δ7)ij + (ε7)ij ,

(16.38)

where we denote

(α7)ij = −1

c2c2

(c2 − w)2

{

Ψij −Dij +

+1

c2vl

[(Dlj + A

∙lj∙

)vi +

(Dli + A

∙li∙

)vj +

1

c2vivjF

l]}∂w

∂t,

(16.39)

(β7)ij =1

c2D×

×

[1

2

(∂vj∂xi

+∂vi∂xj

)

−Dij −1

2c2(Fivj + Fjvi

)− ∗Δlij vl

]

,

(16.40)

(γ7)ij =1

c4viDvl

(Dlj + A

∙lj∙

), (16.41)

(δ7)ij =1

c4vjDvl

(Dli + A

∙li∙

), (16.42)

(ε7)ij =1

c6vivjDvlF

l. (16.43)


−Γkij∂ ln

√−g

∂xk= −

{∗Δkij −

1

c2

[(Dkj + A

∙kj∙

)vi+

+(Dki+A

∙ki∙

)vj+

1

c2vivjF

k

]}(

−1

c2c2

c2−w∂w

∂xk+∂ ln

√h

∂xk

)

=

= (β8)ij + (γ8)ij + (δ8)ij + (ε8)ij ,

(16.44)


where we denote

(β8)ij =1

c2∗Δkij

c2

c2 − w∂w

∂xk− ∗Δkij

∗∂ ln√h

∂xk+1

c2D ∗Δkij vk , (16.45)

(γ8)ij=1

c2vi

[

−1

c2(Dkj+A

∙kj∙

) c2

c2−w∂w

∂xk+(Dkj+A

∙kj∙

)∂ ln√h

∂xk

]

, (16.46)

(δ8)ij=1

c2vj

[

−1

c2(Dki+A

∙ki∙

) c2

c2−w∂w

∂xk+(Dki+A

∙ki∙

)∂ ln√h

∂xk

]

, (16.47)

(ε8)ij =1

c4vivj

(

−1

c2c2

c2 − wF k

∂w

∂xk+ F k

∂ ln√h

∂xk

)

. (16.48)


(α)ij = (α1)ij + (α7)ij , (β)ij=8∑

n=1

(βn)ij ,

(γ)ij=

8∑

n=1

(γn)ij , (δ)ij=

8∑

n=1

(δn)ij , (ε)ij=

8∑

n=1

(εn)ij ,

(16.49)

we can write

Gij = (α)ij + (β)ij + (γ)ij + (δ)ij + (ε)ij . (16.50)

Comparing (α1)ij and (α7)ij , we see that

(α)ij = 0 . (16.51)

We now deduce (β)ij , (γ)ij , (δ)ij , and (ε)ij . Because of (16.5),(16.10), (16.16), (16.21), (16.26), (16.33), (16.40), and (16.45), weobtain the first of the quantities, namely

8∑

n=1

(βn)ij = −1

c2

∗∂Dij

∂t+1

c2

∗∂

∂t

[1

2

(∂vj∂xi

+∂vi∂xj

)

−

−1

2c2(Fivj+Fjvi)

]

+1

c2∗ΔkijFk−

1

c4vl(Dlj+A

∙lj∙

)Fi −

−1

c4vl(Dli + A

∙li∙

)Fj +Hij +

1

c2(Dkj + A

∙kj∙

) ∂vi∂xk

+


+1

c2(Dki + A

∙ki∙

) ∂vj∂xk

+2

c2Dki Djk +

1

c2A∙ki∙Djk +

+1

c2A∙kj∙Dik −

1

c2(Dki + A

∙ki∙

)[1

2

(∂vk∂xj

+∂vj∂xk

)

−

−1

2c2(Fjvk+Fkvj)

]

−1

c2(Dkj +A

∙kj∙

)[1

2

(∂vk∂xi

+∂vi∂xk

)

−

−1

2c2(Fivk + Fkvi)

]

−1

c2c2

c2 − w∂2w

∂xi∂xj−1

c2D×

×

(∂vi∂xj

−1

c2Fivj

)

−1

c2DDij+

1

c2D

[1

2

(∂vj∂xi

+∂vi∂xj

)

−

−1

2c2(Fivj + Fjvi)

]

= Hij +1

c2AijD −

2

c2A∙ki∙ Ajk −

−1

c2

(∗∂Dij

∂t−2Dk

iDjk+DDij

)

−1

c2

{c2

c2−w∂2w

∂xi∂xj−

−∗∂

∂t

[1

2

(∂vj∂xi

+∂vi∂xj

)

−1

2c2(Fivj+Fjvi)

]

−1

c2∗ΔkijFk

}

.

(16.52)

On the other hand,

∗∇i Fj−1

c2FiFj =

∂Fj∂xi

+vi

c2−w∂Fj∂t−∗ΔkijFk−

1

c2FiFj =

=1

c2c2

c2−wFj∂w

∂xi+

c2

c2−w∂2w

∂xi∂xj−

∗∂

∂t

(∂vj∂xi

)

+

+1

c2vi∗∂Fj∂t

−1

c2c2

c2−wFj∂w

∂xi+1

c2Fj

∗∂vi∂t

− ∗ΔkijFk =

=c2

c2−w∂2w

∂xi∂xj−

∗∂

∂t

(∂vj∂xi

−1

c2Fj vi

)

− ∗ΔkijFk ,

(16.53)

so we have

(∗∇i Fj +∗∇j Fi)−

1

c2FiFj =

c2

c2 − w∂2w

∂xi∂xj−

−∂

∂t

[1

2

(∂vj∂xi

+∂vi∂xj

)

−1

2c2(Fivj + Fjvi)

]

− ∗Δkij Fk .

(16.54)

Besides these (see 21.21 of §2.21), we have

Hij = Sij +1

c2(AjiD + AjkD

ki + AikD

kj

). (16.55)


Therefore, we get

(β)ij = Sij −1

c2

[ ∗∂Dij

∂t− 2Dk

iDjk +DDij −Dki Ajk−

− DkjAik + 2A

∙ki∙Ajk +

1

2(∗∇iFj +

∗∇j Fi)−1

c2FiFj

]

.

(16.56)

Because of (16.6), (16.11), (16.17), (16.22), (16.27), (16.34),(16.41), and (16.46), we obtain

8∑

n=1

(γn)ij = −1

c2vi

{

−1

c2Fk(Dkj + A

∙kj∙

)+

+∗∂

∂xk(Dkj + A

∙kj∙

)−1

c4Fj vlF

l+1

c2F k

∂vj∂xk

+1

c2DjkF

k−

−1

c2

[1

2

(∂vk∂xj

+∂vj∂xk

)

−1

2c2(Fjvk + Fkvj)

]

F k−

− ∗Δkjl(Dlk + A

∙lk∙

)−

∗∂D

∂xj+(Dkj + A

∙kj∙

) ∗∂ ln√h

∂xk

}

=

=1

c2vi

[∗∇k

(Dkj + A

∙kj∙

)−

∗∂D

∂xj−2

c2AjkF

k

]

(16.57)

or, finally,

(γ)ij = −1

c2vi

[∗∇k

(hkjD −D

kj

)− ∗∇kA

∙kj∙ +

2

c2AjkF

k

]

. (16.58)

In the same way, using formulae (16.7), (16.12), (16.18), (16.23),(16.28), (16.35), (16.42), and (16.47), we arrive to

(δ)ij = −1

c2vj

[∗∇k

(hkiD −D

ki

)− ∗∇kA

∙ki∙ +

2

c2AikF

k

]

. (16.59)

Formulae (16.8), (16.13), (16.19), (16.24), (16.29), (16.36), (16.43),and (16.48) give

8∑

n=1

(εn)ij =1

c4vivj

(

−1

c2FkF

k+

+∗∂F k

∂xk+ F k

∗∂ ln√h

∂xk+Dl

kDkl + A

∙lk∙A

∙kl∙ +

∗∂D

∂t

)

,

(16.60)


so we obtain

(εn)ij=1

c4vivj

(∗∂D∂t

+DlkD

kl +A

∙lk∙A

∙kl∙ +

∗∇kFk−

1

c2FkF

k

)

. (16.61)

As the final result, we have

Gij = Sij −1

c2

[ ∗∂Dij∂t

− 2DkiDjk +DDij +D

ki Akj +

+ DkjAki − 2A

∙ki∙Akj +

1

2(∗∇i Fj +

∗∇jFi)−1

c2FiFj

]

−

−1

c2vi

[∗∇k

(hkjD −D

kj

)− ∗∇k A

∙kj∙ +

2

c2AjkF

k

]

−

−1

c2vj

[∗∇k

(hkiD −D

ki

)− ∗∇k A

∙ki∙ +

2

c2AikF

k

]

+

+1

c4vivj

(∗∂D∂t

+DlkD

kl +A

∙lk∙A

∙kl∙ +

∗∇k Fk−

1

c2FkF

k

)

,

(16.62)

becauseAkj = −Ajk , Aki = −Aik . (16.63)

§3.17 The Einstein tensor and chr.inv.-quantities

According to §2.3, the quantities

G = gμνGμν = gμνGμν = Gνν , (17.1)

L = c2G00g00

, (17.2)

Mk = −cGk0√g00

, (17.3)

N jk = −c2Gjk (17.4)

are chr.inv.-invariants, a chr.inv.-vector, and a chr.inv.-tensor ofthe 2nd rank, respectively. Using formulae (14.7), (15.7), and (16.62),we are going to deduce detailed formulae for the quantities, and alsofor the Einstein covariant tensor and the invariant G, expressedthrough the quantities.

3.17 The Einstein tensor and chr.inv.-quantities 171


L =∗∂D

∂t+Dl

jDjl + A

∙lj∙A

∙jl∙ +

∗∇j Fj−

1

c2FjF

j . (17.5)


G0i =1

c

(1−

w

c2

)×

×

[∗∇j(hjiD −D

ji

)− ∗∇j A

∙ji∙ +

2

c2AijF

j −1

c2viL

]

.

(17.6)

It is evident that

Gk0 = gk0G00 + gkiG0i = −

1

c3vk(1−

w

c2

)L−

1

c

(1−

w

c2

)×

×

[∗∇j(hkjD −Dkj

)− ∗∇j A

kj +2

c2AkjFj −

1

c2vkL

]

=

= −1

c

(1−

w

c2

)[∗∇j(hkjD −Dkj

)− ∗∇j A

kj +2

c2AkjFj

](17.7)

and, hence (see formula 17.3)

Mk = ∗∇j(hkjD −Dkj

)− ∗∇j A

kj +2

c2AkjFj . (17.8)

Comparing (17.6) and (17.8), we have

G0i =1

c

(1−

w

c2

)(Mi −

1

c2viL

). (17.9)

Comparing (16.62) to (17.8) and (17.5), we have

Gij = Sij −1

c2

[ ∗∂Dij∂t

− 2DkiDjk +DDij +D

ki Akj +

+ DkjAki − 2A

∙ki∙Akj +

1

2

(∗∇i Fj +

∗∇j Fi)−1

c2FiFj

]

−

−1

c2viMj −

1

c2vjMi +

1

c4vivjL .

(17.10)

Because

hjphkq∗∂Dpq

∂t=

∗∂Djk

∂t−Dpq

∗∂(hjphkq

)

∂t=

=∗∂Djk

∂t+ 2DjpDk

p + 2DkqDj

q =∗∂Djk

∂t+ 4DjlDk

l ,

(17.11)


we obtain

Gjk= gjαgkβGαβ = gj0gk0G00+ g

j0gkqG0q + gjpgk0Gp0+

+ g jpgkqGpq =1

c4vjvkL+

1

c2vj(Mk −

1

c2vkL

)+

+1

c2vk(M j −

1

c2vjL

)+ Sjk −

1

c2

[ ∗∂Djk

∂t+

+ 2DjlDkl +DD

jk +DjlA∙kl∙ +DklA

∙jl∙ − 2A

jlA∙kl∙ +

+1

2

(∗∇jF k + ∗∇kF j

)−1

c2F jF k

]

−1

c2vjMk −

−1

c2vkM j +

1

c4vjvkL = Sjk −

1

c2

[ ∗∂Djk

∂t+

+ 2DjlDkl +DD

jk +DjlA∙kl∙ +DklA

∙jl∙ − 2A

jlA∙kl∙ +

+1

2

(∗∇jF k + ∗∇kF j

)−1

c2F jF k

]

(17.12)

and, hence

N jk = −c2Sjk +∗∂Djk

∂t+ 2DjlDk

l +DDjk +DjlA∙kl∙ +

+ DklA∙jl∙ − 2A

jlA∙kl∙ +1

2

(∗∇jF k + ∗∇kF j

)−1

c2F jF k.

(17.13)

Comparing (17.10) and (17.13), we have

Gij = −1

c2

(Nij + viMj + vjMi −

1

c2vivjL

). (17.14)


G =1

c2(L+N) , (17.15)

where the chr.inv.-invariant N is defined as follows

N = hijNij = hijNij = N

jj . (17.16)

Because

hij∗∂Djk

∂t=

∗∂Dki

∂t−Djk

∗∂hij∂t

=∗∂Dk

i

∂t− 2DijD

jk, (17.17)

3.17 The Einstein tensor and chr.inv.-quantities 173

we obtain

Nki = −c

2Ski +∗∂Dk

i

∂t+DDk

i +DliA

∙kl∙ +D

kl A

l∙∙i−

− 2A∙li∙A∙kl∙ +

1

2

(∗∇i F

k + ∗∇kFi)−1

c2FiF

k.

(17.18)


N = −c2S +∗∂D

∂t+D2 − 2A∙lj∙A

∙jl∙ +

∗∇j Fj −

1

c2FjF

j , (17.19)

G=−S+1

c2

(

2∗∂D

∂t+D2+Dl

jDjl −A

∙lj∙A

∙jl∙+2

∗∇jFj−

2

c2FjF

j

)

. (17.20)

We now collect the formulae we have obtained. Using formula(20.18) of §2.20, and introducing the concise notation

∗∇i =∗∇i −

1

c2Fi ,

∗∇i = ∗∇i −1

c2F i, (17.21)

we have∗

L =∗∂D

∂t+Dl

jDjl + A

∙lj∙A

∙jl∙ +

∗∇j Fj

Mk = ∗∇j(hkjD −Dkj

)− ∗∇j A

kj +2

c2AkjFj

Nki = −c

2Ski +∗∂Dk

i

∂t+DDk

i +DliA

∙kl∙ +D

kl A

l∙∙i −


1

2

(∗∇i F

k + ∗∇kFi)

, (17.22)

N = −c2S +∗∂D

∂t+D2 − 2A∙lj∙A

∙jl∙ +

∗∇j Fj , (17.23)

G = −S+1

c2

(

2∗∂D

∂t+D2+Dl

jDjl −A

∙lj∙A

∙jl∙+2

∗∇jFj

)

, (17.24)

∗Zelmanov regularly called tensor quantities, contracted with the chr.inv.-operator ∗∇ (17.21) in one index, their physical chronometrically invariantdivergence. As a matter of fact, the physical chr.inv.-divergence and regular chr.inv.-divergence (the contraction with ∗∇i in one index, see §2.14) are not the same. Forinstance, ∗∇iQi and ∗∇iQij are the regular chr.inv.-divergences of the chr.inv.-vector Qi and the chr.inv.-tensor of the 2nd rank Qij , while ∗∇iQi and ∗∇iQij arethe physical chr.inv.-divergences of the same quantities. — Editor’s comment. D. R.


G00 =1

c2

(1−

w

c2

)2L

G0i =1

c

(1−

w

c2

)(Mi −

1

c2viL)

Gij = −1

c2

(Nij + viMj + vjMi −

1

c2vivjL

)

, (17.25)

G =1

c2(L+N) . (17.26)

§3.18 The law of gravitation. Its time covariant equation

Let us take the equations of gravitation in the form

Gμν = −κ(Tμν −

1

2gμν T

)+ Λ gμν (18.1)

and consider the equations with μ, ν =0, i. e. the time equation

G00 = −κ(T00 −

1

2g00 T

)+ Λ g00 . (18.2)


T00 −1

2g00 T =

1

2

(1−

w

c2

)2(ρ+

1

c2U). (18.3)

Because of (17.25), the initial equation (18.2) becomes

L = −κ

2

(ρc2 + U

)+ Λc2. (18.4)

§3.19 The law of gravitation. The mixed covariant equations

We next consider the mixed (space-time) equations of gravitation,which are the equations (18.1) with μ=0 and ν=1, 2, 3

G0i = −κ(T0i −

1

2g0i T

)+ Λ g0i . (19.1)


T0i −1

2g0i T = −

1

c

(1−

w

c2

)[Ji +

vi2c2(ρc2 + U

)], (19.2)

then, taking (17.25) into account, the equations (19.1) give

Mi −1

c2viL = κ

[Ji +

vi2c2(ρc2 + U

)]− Λvi (19.3)

3.20 The law of gravitation. The spatial covariant equations 175

or, in the other form

Mi − κJi =1

c2vi

[L+

κ

2

(ρc2 + U

)− Λc2

]. (19.4)

Because formula (19.4) holds for any choice of the time coord-inate, we can apply the method to vary the potentials. Setting allvalues of vi to zero, we obtain

Mi = κJi . (19.5)

Formula (19.5) retains its form for any choice of the time coord-inate, because of its chr.inv.-tensor (namely — chr.inv.-vector) na-ture. Therefore (19.4) leads to, besides (19.5), the equality

L = −κ

2

(ρc2 + U

)+ Λc2, (19.6)

which coincides with early obtained formula (18.4) and is a chr.inv.-tensor as well.

§3.20 The law of gravitation. The spatial covariant equations

We now consider the spatial equations of gravitation, which areequations (18.1) with μ, ν =1, 2, 3

Gij = −κ(Tij −

1

2gij T

)+ Λgij . (20.1)


Tij −1

2gij T =

1

c2

[

Uij −1

2hijU +

1

2hijρc

2+

+ viJj + vjJi +1

2c2vivj

(ρc2 + U

)]

.

(20.2)

Because of (17.25) and (20.2), the initial formula (20.1) can bewritten as follows

Nij+viMj+vjMi−1

c2vivjL = κ

[

Uij−1

2hijU+

1

2hijρc

2+

+ viJj + vjJi +1

2c2vivj

(ρc2+U

)]

+ Λc2(hij −

1

c2vivj

) (20.3)



Nij−κ(Uij−

1

2hijU+

1

2hijρc

2)−Λc2hij+ vi (Mj−κJj) +

+ vj (Mi−κJi)−1

c2vivj

[L+

κ

2

(ρc2 + U

)−Λc2

]=0 .

(20.4)

Because (20.4) is true for any choice of the time coordinate, wecan use once again the method to vary the potentials. Setting allthe vk to zero, we obtain

Nij = κ(Uij −

1

2hijU +

1

2hijρc

2)+ Λc2hij . (20.5)

The equality we have obtained, as well as any chr.inv.-tensorequality, is valid for any choice of the time coordinate. Hence

vi (Mj−κJj)+ vj (Mi−κJi)−1

c2vivj

[L+

κ

2

(ρc2+U

)−Λc2

]=0 (20.6)

is also true for any choice of the time coordinate. Next, we sup-pose that

v1 6= 0 , v2 = v3 = 0 , (20.7)

then (20.6) takes the form (after v1 has been excluded)

2 (M1 − κJ1)−1

c2

[L+

κ

2

(ρc2 + U

)− Λc2

]= 0

M2 − κJ2 = 0 , M3 − κJ3 = 0

. (20.8)

Because (20.8) is true for any v1 6=0, we obtain the equalities

Mi = κJi , (20.9)

L = −κ

2

(ρc2 + U

)+ Λc2. (20.10)

The equalities retain their form for any choice of the time coord-inate because of their chr.inv.-tensor nature. It is evident that theequalities coincide with formulae (19.5) and (18.4).

§3.21 The scalar equation of gravitation

We finally consider the scalar equation

G = κT + 4Λ , (21.1)

3.22 The equations of gravitation: the 1st chr.inv.-form 177

which is a consequence of the tensor equations of gravitation. So,because of (17.26) and (10.44), equation (21.1) takes the form

L+N = κ(ρc2 − U

)+ 4Λc2 (21.2)


L+N = κρ0c2 + 4Λc2. (21.3)

§3.22 The first chr.inv.-tensor form of the equations of gravi-tation

We have obtained the system of equations (18.4), (19.5), and (20.5).Actually, this system is a chr.inv.-tensor form for the equations ofgravitation. We will refer to the system as the first chr.inv.-formof the equations of gravitation. Let us write the system in the form

L = −κ

2

(ρc2 + U

)+ Λc2

Mk = κJ k

N ik = κ(U ik −

1

2hikU +

1

2hikρc2

)+ Λc2hik

. (22.1)

It is easy to see that the first of the equations (the chr.inv.-invariant) is equivalent to the equation

G00 = −κ(T00 −

1

2g00 T

)+ Λ g00 , (22.2)

the second (chr.inv.-vector) equation is equivalent to

Gk0 = −κ(T k0 −

1

2gk0 T

)+ Λ gk0 , (22.3)

which coincides with the equations

Gk0 = −κTk0 , (22.4)

and finally, the third (chr.inv.-tensor) equation is equivalent to theequations

Gik = −κ(T ik −

1

2gik T

)+ Λ gik. (22.5)

We re-write the system (22.1) in component form, having low-ered the index i in the tensor equation, and also using (17.22).So, we have

∗∂D

∂t+Dl

jDjl +A

∙lj∙A

∙jl∙ +

∗∇jFj = −

κ

2

(ρc2+U

)+ Λc2, (22.6)


∗∇j(hkjD −Dkj

)− ∗∇jA

kj +2

c2AkjFj = κJ k, (22.7)

− c2Ski +∗∂Dk

i

∂t+DDk

i +DliA

∙kl∙ +D

kl A

l∙∙i − 2A

∙li∙A

∙kl∙ +

+1

2

(∗∇iF

k+∗∇kFi)=κ(Uki −

1

2hki U+

1

2hkiρc

2)+Λc2hki .

(22.8)

Because of (9.2) and also (16.3) of §2.16, we obtain

Ail = εjilΩj , Alk = εqlkΩq , (22.9)

A∙li∙A∙kl∙ = AilA

lk = εjil εqlkΩjΩq = −εjil ε

qklΩq Ωj =

= −(hqj h

ki − h

qi h

kj

)Ωq Ω

j = −hki ΩjΩj +ΩiΩ

k,(22.10)

A∙lj∙A∙jl∙ = −2ΩjΩ

j . (22.11)

Because of (9.2) and also (16.4), (14.18), (16.15) obtained inChapter 2, we get

∗∇jAkj=∗∇j

(εpkjΩp

)= εpkj ∗∇jΩp= ε

kjp∗∇jΩp=∗rk(Ω) . (22.12)

Hence, because of (9.2), we obtain

AkjFj = εpkj ΩpFj = −εpjk ΩpFj = −ε

jlk Ωj Fl , (22.13)

DliA

∙kl∙ +D

klA

l∙∙i = DilA

lk +DlkAli = εjlkDilΩj + εjliDlkΩj . (22.14)

Using (22.10–22.14) and also (19.18), (19.19) of §2.19, we can re-write equations (22.6), (22.7), and (22.8), respectively, in the forms

∗∂D

∂t+Dl

jDjl −2ΩjΩ

j+ ∗∇j Fj = −

κ

2

(ρc2+U

)+Λc2, (22.15)

Rk (∗ω) = ∗rk (Ω)−2

c2εjlkΩjFl + κJ

k, (22.16)

− c2Ski +∗∂Dk

i

∂t+DDk

i + εjlkDilΩj + εjliD

lkΩj−

− 2(ΩiΩ

k − hki ΩjΩj)+1

2

(∗∇i F

k + ∗∇kFi)=

= κ(Uki −

1

2hki U +

1

2hki ρc

2)+ Λc2hki .

(22.17)

Equation (21.1) can be (because of 17.24 and 10.44) re-written

3.23 The equations of gravitation: the 2nd chr.inv.-form 179

as follows

−c2S+2∗∂D

∂t+D2+Dl

jDjl−A

∙lj∙A

∙jl∙+2

∗∇jFj=κ

(ρc2−U

)+4Λc2. (22.18)

Using (22.11) again, we put equation (22.18) into the form

−c2S+2∗∂D

∂t+D2+Dl

jDjl+2ΩjΩ

j+2∗∇jFj=κ

(ρc2−U

)+4Λc2. (22.19)

§3.23 The second chr.inv.-tensor form of the equations of gravi-tation

We consider next the system of equations

G00 −1

2g00G = −κT00 − Λg00 , (23.1)

Gk0 −1

2gk0 G = −κT

k0 − Λg

k0 , (23.2)

where the second coincides with (22.4)

Gk0 = −κTk0 , (23.3)

and also the equation

Gjk −1

2gjkG = −κT jk − Λ gjk. (23.4)


G00 −1

2g00G =

1

2c2

(1−

w

c2

)2(L−N) . (23.5)

For this reason, and because of (10.38), the equation (23.1) isequivalent to

1

2(N − L) = κρc2 + Λc2. (23.6)

As a result of (17.4) and (17.26), we have

Gjk −1

2gjkG =

1

c2

[1

2hjk(L+N)−N jk

]

, (23.7)

and the third equation (23.4) is, as a result of (10.40), equivalent to

N jk −1

2hjk (L+N) = κU jk − Λc2hjk. (23.8)

As a result, it is easy to see that the system of equations (23.1),


(23.2), and (23.4) is equivalent to the system of equations (22.2),(22.3), and (22.5).

We can see directly that the system of equations

1

2(N − L) = κρc2 + Λc2

Mk = κJ k

N jk −1

2hjk (L+N) = κU jk − Λc2hjk

(23.9)

is equivalent to the system (22.1), which is the first chr.inv.-tensorform of the equations of gravitation. For this reason, we will referto the system (23.9) as the second chr.inv.-tensor form of the equa-tions of gravitation.

Using (17.22) and (17.23),we re-write equations (23.6) and (23.8)of the system (23.9) in component form (having lowered the indexi in 23.8). As a result we obtain

1

2

(−c2S +D2 −Dl

jDjl − 3A

∙lj∙A

∙jl∙

)= κρc2 + Λc2, (23.10)

− c2(Ski −

1

2hki S

)+

∗∂Dki

∂t+DDk

i +DliA

∙kl∙ +D

kl A

l∙∙i −


1

2

(∗∇i F

k + ∗∇kFi)− hki

[ ∗∂D∂t

+

+1

2

(D2 +Dl

jDjl − A

∙lj∙A

∙jl∙

)+ ∗∇j F

j

]

= κUki − Λc2hki .

(23.11)

Using (22.11), (22.14), (22.10), and also (21.29), (21.35) of §2.21,we put equations (23.10) and (23.11), respectively, into the forms

c2Z +1

2

(D2 −Dl

jDjl

)+ 3ΩjΩ

j = κρc2 + Λc2, (23.12)

− c2Zki +∗∂Dk

i

∂t+DDk

i + εjlkDilΩj + εjliD

lkΩj−

− 2ΩiΩk +

1

2

(∗∇i F

k + ∗∇kFi)− hki

[ ∗∂D∂t

+

+1

2

(D2 +Dl

jDjl

)− ΩjΩ

j + ∗∇j Fj

]

= κUki − Λc2hki .

(23.13)

We also re-write (22.19), using (21.35) of §2.21. So, we obtain

c2Z+∗∂D

∂t+1

2

(D2+Dl

jDjl

)+ΩjΩ

j+∗∇jFj=

κ

2

(ρc2−U

)+2Λc2. (23.14)

3.24 The structure of the equations of gravitation 181

§3.24 The structure of the equations of gravitation

Examining the chr.inv.-tensor forms of the equations of gravitation,we see that some quantities appear only in combination with otherquantities. A peculiarity of the combinations is that in eliminatingone of the quantities from the equations, we eliminate the entirecombination. The combinations are

ϕki =∗∂Dk

i

∂t+1

2

(∗∇i F

k + ∗∇kFi)− κ

(Uki −

1

2hki U

), (24.1)

ϕ =∗∂D

∂t+ ∗∇j F

j +κ

2U , (24.2)

σki = −c2Ski +DD

ki , (24.3)

σ = −c2S +D2, (24.4)

ωki = DliA

∙kl∙ +D

kl A

l∙∙i = εjlkDilΩj + εjliD

lkΩj . (24.5)

We introduce the notation∗

αki = −A∙li∙A

∙kl∙ = −ΩiΩ

k + hki ΩjΩj , (24.6)

α = −A∙lj∙A∙jl∙ = 2ΩjΩ

j , (24.7)

δ = DljD

jl , (24.8)

θk = ∗∇j(hkjD −Dkj

)− ∗∇jA

kj +2

c2AkjFj − κJ

k =

= Rk (∗ω)− ∗rk (Ω) +2

c2εjlkΩjFl − κJ

k.(24.9)

It is also evident that

ϕjj = ϕ , σ

jj = σ , ω

jj = 0 , α

jj = α . (24.10)

The equations of gravitation take the following form. The firstchr.inv.-form is

α+ Λc2 = δ + ϕ+κ

2ρc2

θk = 0

2αki + σki + ϕ

ki + ω

ki =

κ

2hki ρc

2 + Λc2hki

, (24.11)

∗See the definition of chr.inv.rotor ∗rk (Ω)= εqpk∗∇qΩp= ∗∇jakj in §2.16. —Editor’s comment. D. R.


the second chr.inv.-form is

3α+ σ = δ + 2κρc2 + 2Λc2

θk = 0(2αki −

1

2hki α

)+(σki −

1

2hki σ

)+(ϕki − h

ki ϕ)+

+ ωki + Λc2hki =

1

2hki δ

, (24.12)

the scalar equation is

α+ δ + σ + 2ϕ = κρc2 + 4Λc2. (24.13)

Let us consider the variable chr.inv.-invariants ρ, α, δ, ϕ, σ,which are contained in the equations of gravitation. From the phys-ical perspective, we assume that

ρ > 0 . (24.14)

We next introduce spatial coordinates that are locally Cartesianand orthogonal at the given world-point. Then we have

Ωj = Ωj , Dk

i = Dik (24.15)

and, hence, it is easy to see that,

α > 0 , (24.16)

δ > 0 . (24.17)

Because α and δ are sub-invariants, the conditions (24.16) and(24.17) hold in any coordinates.

Inspecting the first equation of the system (24.11), we see that,generally speaking,

ϕ T 0 . (24.18)

Similarly, inspection of the first equation of the system (24.12),we see that, generally speaking,

σ T 0 . (24.19)

♦

Chapter 4

NUMEROUS COSMOLOGICAL CONSEQUENCES

§4.1 Locally-accompanying frames of reference

We are going to consider substance as continuous media. We canintroduce a locally accompanying reference frame at any point Aof a substance (the point A must not be a break-point for the sub-stance’s velocity with respect to the initial reference frame we havechosen arbitrarily). We introduce the reference frame as follows:

A reference frame, which locally accompanies a substance at agiven point A, is such that at this point A of the substance forthe time interval t we are considering, (1) the frame is locally-stationary, and (2) the frame does not rotate with respect tothe space.

We assume that ∗ui is the chr.inv.-velocity of the substance withrespect to the reference frame which accompanies the substancelocally at the point A. Then, at the point A, we have

(∗ui)A≡ 0 , (1.1)

(∗ωk

)A≡ 0 . (1.2)

Because formula (10.6) of Chapter 2 gives (10.8), (17.12), and(17.13), taking (1.1) and (1.2) into account, we obtain

(ui)A≡ 0 , (1.3)

(ωk)A≡ 0 . (1.4)

§4.2 Accompanying frames of reference

In accordance with §1.10 and §1.19, we will refer to the referenceframe which moves in company with the substance at any world-point of the given four-dimensional volume, as the accompanying

184 Chapter 4 Numerous Cosmological Consequences

reference frame. If ∗ui is the chr.inv.-velocity of the substance withrespect to the given reference frame, then, in this accompanyingreference frame, we have

∗ui ≡ 0 (2.1)

and, henceui ≡ 0 . (2.2)

It is clear that in any volume, where the substance’s velocityhas no break-points (with respect to an arbitrary reference frame),we can introduce the accompanying reference frame. Naturally, letus assume that components ui of the substance’s velocity are finite,simple, and continuous functions∗ of their arguments x0, x1, x2, x3

in a four-dimensional volume Q of a coordinate frame S. Then thesystem of functions

xi′= xi

′(x0, x1, x2, x3) , i = 1, 2, 3 (2.3)

exist, where the functions are finite, simple, continuous, and dif-ferentiable in the volume Q, and satisfy the equations

∂xi′

∂x0+∂xi

′

∂xjuj = 0 (2.4)

in this volume. We introduce a coordinate frame S′, which is linkedto the coordinate frame S by the transformations

x0′= x0

xi′= xi

′(x0, x1, x2, x3)

}

. (2.5)

Let us find the velocity vi of the motion of the system S′ in thesystem S. Because

dx0 = dx0′

dxi =∂xi

∂x0′dx0

′+∂xi

∂xj′ dx

j ′

, (2.6)

we have

vi =∂xi

∂x0′. (2.7)

∗In the §4.1 and §4.2 above, we did not mention that the velocity is finite andunique, because the properties were implicit from physical considerations.

4.3 The kinds of matter. Their motions 185

On the other hand, because

∂xi′

∂x0′≡ 0 , (2.8)

we have∂xi

′

∂x0+∂xi

′

∂xj∂xj

∂x0′= 0 . (2.9)

Therefore, we obtain

∂xi′

∂x0+∂xi

′

∂xjvj = 0 . (2.10)

Subtracting (2.4) from (2.10) term-by-term, we obtain

∂xi′

∂xj(vj − uj

)= 0 (2.11)

and because∂ (x1

′, x2

′, x3

′)

∂ (x1, x2, x3)6≡ 0 , (2.12)

we getvi ≡ ui . (2.13)

So the coordinate frame S′ moves in company with the sub-stance, and hence, S′ is of the reference frame which accompaniesthe volume Q.

§4.3 The kinds of matter. Their motions

We assume that the four-dimensional volume Q we are consideringis filled with matter which satisfies the following conditions:

1. The matter is a continuously distributed substance — a con-tinuous medium, which has no pressure or stresses, is free ofheat flux, and has positive density;

2. At any world-point of the volume Q, the components ui ofthe substance’s velocity with respect to an arbitrary referenceframe (x0,x1,x2,x3), real numerical values of coordinates ofwhich cover the neighbourhood of this world-point, are finite,simple, and differentiable functions of the time coordinate andthe spatial coordinates.

The first condition implies that the matter can be considered asan ideal fluid, free of heat flux, of positive density ρ00 and zero


pressure p0. So, we have

Tμν =(ρ00 +

p0c2

)dxμ

ds

dxν

ds−p0c2gμν , ρ00 > 0 , p0 = 0 (3.1)


Tμν = ρ00dxμ

ds

dxν

ds, ρ00 > 0 . (3.2)

It is evident that

T00 = ρ00dx0ds

dx0ds

, (3.3)

T i0 = ρ00dx0ds

dxi

ds, (3.4)

T ik = ρ00dxi

ds

dxk

ds. (3.5)

The second condition, taking the results of §4.2 into account,implies that we can cover the volume Q point-by-point by theaccompanying coordinate frames. In other words, we can introducethe accompanying reference frame throughout the volume Q.

§4.4 Matter in the accompanying frame of reference

We assume that matter∗ is linked to the accompanying referenceframe. For any point-mass of the substance in this reference frame

dxi = 0 , i = 1, 2, 3 , (4.1)

is true, hencedx0 = g00dx

0, (4.2)

ds2 = g00dx0dx0. (4.3)

Therefore, because of (3.3) and (3.5), we obtain

ρ = ρ00 , (4.4)

J i = 0 , (4.5)

U ik = 0 . (4.6)

It is evident that the equalities (4.4–4.6) are essentially linkedto our suppositions about the properties of matter. On the contrary,

∗A substance and radiations. — Editor’s comment. D. R.

4.4 Matter in the accompanying frame of reference 187

speculations in this section are independent of the suppositionsgiven for the 1st condition of §4.3; the speculations are linked onlyto the 2nd condition, containing only suppositions about motions ofthe substance. The substance is at rest with respect to the accom-panying reference frame by definition of such reference frames. Inthe neighbourhood of every point of the accompanying space, thesubstance, generally speaking, moves with respect to the referenceframe locally accompanying the substance at the point. At everypoint of the accompanying space we can introduce: (1) the chr.inv.-tensors (covariant, mixed and contravariant) and the chr.inv.-invariant of the rate of deformations of the substance in respectof the reference frame, which is locally accompanying in this point;(2) the chr.inv.-rotor of the chr.inv.-vector of the angular velocitiesof rotations of the substance with respect to the same locally ac-companying reference frame. Because the accompanying spacemoves with the substance, the aforementioned quantities coincidewith the chr.inv.-tensors and the chr.inv.-invariant of the rate ofdeformations of the accompanying space and, respectively, the chr.inv.-rotor of the chr.inv.-vector of the angular velocities of theaccompanying space with respect to the reference frame locallyaccompanying at this point. At the same time the locally accompa-nying reference frame is of the locally-stationary reference frames.For this reason, we can identify the aforementioned quantities withthe quantities Dik, Dk

i , Dik, D, and Ri(∗ω)= ∗∇j

(hijD−Dij

), re-

spectively∗.Thus, the quantities Dik, Dk

i , Dik, D, and Ri(∗ω), characterizing

the rate of deformations and the velocities of relative rotationsof elements of the accompanying space, actually characterize thesame for an element of the substance. In particular, the chr.inv.-invariant of the rate of relative expansions of an elementary volumeof the space, namely — the quantity D, gives the rate of relativeexpansions of the element of the substance at any given point.

In addition to the above, we can also say that, in the accompany-ing reference frame, relative motions of elements of the substancecharacterize themselves by the fact that the quantities hik (hence,

∗As mentioned above, it can be a set of locally-stationary reference frames atany given point. We saw that the arbitrary rule for choosing the locally-stationaryreference systems does not affect the aforementioned equations containing thequantities Dik, Dk

i , Dik, D, and Ri(∗ω). Now, as seen from this section,we specialize this choice of locally-stationary reference frames, taking locallyaccompanying reference frames instead of them.


also the quantities hik and h) are dependent on the time coordinate.In particular, transformations of the volume of the substance’selement are described by a function of

√h from t (see formula

12.7 of §2.12).

§4.5 The cosmological equations of gravitation

We are going to consider the system of the equations of gravita-tion. This system in the case (3.2) can be considered as a systemof 10 equations with respect to 10 unknowns: one quantity ρ, threeuk and 6 of 10 independent quantities gμν .∗ At the same time, theequations can be considered as equations that define ρ from 9 ofthe components gμν when uk and one other component of gμν aregiven. Introducing the accompanying reference frame, we applythis approach to the equations of gravitation. Therefore, the systemof the equations of gravitation in the accompanying reference framecan be considered, for instance, as a system, which define the valueof ρ, the 3 value of vi, and the 6 values of hik (when w is given bya special choice of the time coordinate).

So, we can write the equations of gravitation in the accompa-nying reference frame as formulae (24.11–24.13) of §3.24: the firstchr.inv.-form

α+ Λc2 = δ + ϕ+κ

2ρc2

θk = 0

2αki + σki + ϕ

ki + ω

ki =

κ

2hki ρc

2 + Λc2hki

, (5.1)

the second chr.inv.-form

3α+ σ = δ + 2κρc2 + 2Λc2

θk = 0(2αki −

1

2hki α

)+(σki −

1

2hki σ

)+(ϕki − h

ki ϕ)+

+ ωki + Λc2hki =

1

2hki δ

, (5.2)

the scalar equation of gravitation

α+ δ + σ + 2ϕ = κρc2 + 4Λc2. (5.3)

∗We can define the other 4 components of gμν by a specific choice of coordinateframe. For instance, see [64], p. 237.

4.5 The cosmological equations of gravitation 189

Besides these, formulae (24.3–24.8) of §3.24 remain unchanged,while formulae (24.1), (24.2), and (24.9) will be simplified (becauseof formulae 4.5 and 4.6 of §4.4) as follows

ϕki =∗∂Dk

i

∂t+1

2

(∗∇i F

k + ∗∇kFi), (5.4)

ϕ =∗∂D

∂t+ ∗∇j F

j , (5.5)

σki = −c2Ski +DD

ki , (5.6)

σ = −c2S +D2, (5.7)

ωki = DliA

∙kl∙ +D

kl A

l∙∙i = εjlkDilΩj + εjliD

lkΩj , (5.8)

αki = −A∙li∙A

∙kl∙ = −ΩiΩ

k + hki ΩjΩj , (5.9)

α = −A∙lj∙A∙jl∙ = 2ΩjΩ

j , (5.10)

δ = DljD

jl , (5.11)

θk = ∗∇j(hkjD −Dkj

)− ∗∇jA

kj +2

c2AkjFj =

= Rk (∗ω)− ∗rk (Ω) +2

c2εjlkΩjFl .

(5.12)

It is also evident that (24.15) and (24.17–24.19) of §3.24 retaintheir power, whilst the equality sign in (24.14) disappears, so

ρ > 0 , (5.13)

α > 0 , (5.14)

δ > 0 , (5.15)

ϕ T 0 , (5.16)

σ T 0 . (5.17)

Thus, we see that the equations (5.1–5.3), because of (5.4–5.12),create a link between the following quantities:

• Quantities which characterize the state of the matter;

• Quantities which characterize the evolution of the accompa-nying space;

• Quantities which characterize the geometric properties of thespace.


In other words, equations (5.1–5.3) are cosmological equations.In accordance with §1.17, they are the cosmological equations ofgravitation. If we use the equations to find ρ, vi, and hik, we willthen need to add 18 other equations to them. The additional eq-uations define Dik (6 components), Zik (6 components), Ωi (3 com-ponents), and Fi (3 components) as functions of w, vi, and hik. Sowe will need to consider the whole system of 28 equations withrespect to the same number of unknown functions.

§4.6 The cosmological equations of energy

The equations of the law of energy (see formulae 12.9 and 12.11 of§3.12), because of (4.3) and (4.6), take the form

∗∂ρ

∂t+ ρD = 0 , (6.1)

F kρ = 0 . (6.2)

Formula (6.2), because of (5.13), leads to

Fi = 0 . (6.3)

In accordance with §1.17, equations (6.1) and (6.2)∗ are thecosmological equations of energy.

It is known that the equations of the law of energy are conse-quences of the equations of the law of gravitation, in the sense thatthey are derived from the four identities between the left sidesof the equations of gravitation. For this reason we can use theequations of energy instead of the four aforementioned identities.

§4.7 The main form of the cosmological equations

Because of the third of equations (5.1), we obtain

2α+ σ + ϕ =3κ

2ρc2 + 3Λc2 (7.1)

and so

2(αki −

1

3hkiα)+(σki −

1

3hki σ)+(ϕki −

1

3hkiϕ)+ ωki = 0 . (7.2)

Adding the first of equations (5.1) and the equation (7.1) term-

∗Or formula (6.3) instead of (6.2).

4.7 The main form of the cosmological equations 191

by-term, we obtain the first of equations (5.2). The last, in its con-nection with the first of equations (5.1), gives (7.1). Hence, equation(7.1), in connection with (7.2), leads to the third of equations (5.1).Therefore, the system

α+ Λc2 = δ + ϕ+κ

2ρc2

3α+ σ = δ + 2κρc2 + 2Λc2

2(αki −

1

3hkiα)+(σki −

1

3hki σ)+(ϕki −

1

3hkiϕ)+ ωki = 0

θk = 0

(7.3)

is equivalent to the system (5.1) and therefore to the system (5.2). Iteasy to see that the third of equations (7.3) gives only 5 independentrelations, because of its symmetry and becoming zero under itsreduction.

Because

ε∙lkj hil + εjlihlk = ε∙∙kji∙ + ε

∙k∙j∙i = hkq

(εjiq + εjqi

)= 0 , (7.4)

we can write (5.8) in the form

ωki = εjlkΩj

(Dil −

1

3hilD

)+ εjliΩ

j(Dlk −

1

3hlkD

). (7.5)

Using (21.33) and (21.35) of §2.21, instead of (5.6) and (5.7), wecan write

σki = −c2(Zki − h

kiZ)+DDk

i , (7.6)

σ = 2c2Z +D2. (7.7)


Π = DljD

jl −

1

3D2, (7.8)

we re-write (5.11) in the form

δ =1

3D2 +Π . (7.9)

Using equation (6.3), equalities (5.4), (5.5), (5.12) become themore simple

ϕki =∗∂Dk

i

∂t, (7.10)


ϕ =∗∂D

∂t, (7.11)

θk = ∗∇j(hkjD−Dkj

)−∗∇jA

kj = Rk(∗ω)−∗rk(Ω) . (5.12)

Let us write equations (6.1), (6.3) and the system (7.3), takingequalities (5.9), (5.10), (7.5–7.7), (7.9–7.12) into account,

∗∂ρ

∂t+ ρD = 0 , (7.13)

Fi = 0 , (7.14)

∗∂D

∂t+1

3D2 +Π− 2ΩjΩ

j = −κ

2ρc2 + Λc2, (7.15)

1

3D2 −

1

2Π + 3ΩjΩ

j + c2Z = κρc2 + Λc2, (7.16)

∗∂

∂t

(Dki −

1

3hkiD

)+D

(Dki −

1

3hkiD

)+

+ εjlkΩj

(Dil −

1

3hilD

)+ εjli Ω

j(Dlk −

1

3hlkD

)=

= 2(ΩiΩ

k −1

3hki ΩjΩ

j)+ c2

(Zki −

1

3hkiZ

),

(7.17)

Rk(∗ω)= ∗rk

(Ω), (7.18)

where the equality (7.18) can be written in the form

∗∇j(hkjD −Dkj

)= ∗∇jA

kj . (7.19)

We take equations (7.13–7.17) and (7.18) as the main chr.inv.-form of the cosmological equations.

§4.8 Free fall. The primary coordinate of time

Equality (7.14) implies that the substance falls freely in the field ofgravitational inertial forces. This is self-evident because there areno other forces under the conditions (4.5) and (4.6). Equality (7.14)leads to numerous consequences∗, one of which will be considerednow — that there is a possibility of choosing the primary coordinateof time at any point of the space.

∗We used the equality, numbered as (6.3), earlier in §4.7.

4.8 Free fall. The primary coordinate of time 193

We assume that condition (7.14) holds everywhere in a four-dimensional volume Q. We introduce a coordinate of time such thatthe condition

w = 0 (8.1)

holds throughout the volume. Then, because of (7.14), we obtain

∂vi∂t

= 0 . (8.2)

The formulae (8.1) and (8.2) lead to

Y = 0 , (8.3)

Φi = 0 (8.4)

in the volume Q (see formulae 23.18 and 23.28 of §2.23). So, numer-ical values of 5 quantities (w, Y , Φi) of the 14 locally independentquantities we considered in §2.20 are defined throughout the vol-ume Q. Next, we need to find numerical values of the other 9quantities (vi and Xik) at any point of the volume Q. We assumethat, at a space point A at a moment t= t0, we have

vi = 0 , (8.5)

Xik = 0 . (8.6)

Because (8.2) holds throughout the volume Q, the condition (8.5)remains unchanged at the point A for all numerical values of t inthe said four-dimensional volume.

Owing to (8.5), formula (8.6) can be written as follows (seeformula 23.31 of §2.23)

∂vk∂xi

+∂vi∂xk

= 0 . (8.7)

Because we have

∂

∂t

(∂vq∂xp

)

=∂

∂xp

(∂vq∂t

)

(8.8)

and (8.2) holds throughout the volume Q, we have everywhere inthat volume

∂

∂t

(∂vk∂xi

+∂vi∂xk

)

= 0 . (8.9)

Hence, formula (8.7) remains unchanged at the point A for allnumerical values of t in the volume Q. It is evident that (8.6) also


remains unchanged at the point A for all values of t in the vol-ume Q.

So, if the conditions (7.14) are true in the volume Q, then wehave the possibility of choosing, at any point A of this volume,time coordinates which permit conditions (8.1) and (8.3–8.6) at thepoint A for all t inside the four-dimensional volume. In accordancewith §2.22, the spatial section, corresponding to this choice of timecoordinates, is a maximally orthogonal one (at the point A for allvalues of t in the volume Q), so that

Kkjin = Skjin (8.10)

and henceHik = Sik , Cik = Zik . (8.11)

If the conditionAik = 0 , (8.12)

aside of the condition (8.1), is also true in the volume Q, then wecan, in accordance with §2.7, introduce the cosmic universal time(see §1.2) — time coordinates by which the equalities (8.1), (8.5),and hence (8.3), (8.4), (8.6), (8.10), (8.11) hold everywhere in thevolume Q.

§4.9 Characteristics of anisotropy

We consider next the problem of anisotropy of volume elements,in both the mechanical and the geometrical senses. Let us firstconsider the problem of the mathematical characteristics of the an-isotropy of that mechanical geometrical factor, which is describedby a symmetric sub-tensor of the 2nd rank Bik. If this factor isspatially isotropic at a world-point, then, in locally Cartesian spatialcoordinates at this point, we have

Bik = Bki = Bik =

1

3B , i = k

0 , i 6= k

, (9.1)

where we denoteB = B

jj . (9.2)

In arbitrary coordinates, we have

Bik =1

3hikB , Bki =

1

3hkiB , Bik =

1

3hikB . (9.3)

4.10 Absolute rotations 195

Next, we introduce the sub-invariant

Γ =(B lj −

1

3hljB

)(Bjl −

1

3hjlB)= B l

jBjl −

1

3B2 . (9.4)

In locally Cartesian coordinates the sub-invariant is equal tothe sum of the squares of the quantities B

jl −

13 h

jlB. So the sub-

invariant is zero under the isotropy conditions (9.3), and it is po-sitive if conditions (9.3) are violated. For these reasons, we canidentify the introduced sub-invariant Γ as the quantity of the spa-tial anisotropy.

It follows from what has been said that, in particular, the chr.inv.-invariant Π=D l

jDjl −

13D

2 (7.8) has a nonnegative numericalvalue, and the chr.inv.-invariant characterizes the anisotropy of de-formations of the volume element (for instance, compare this resultwith [58], p. 612).

Let us consider the case of

Bik = βi βk , (9.5)

where βj is a sub-vector. Then

B = βj βj , Γ =

2

3

(βj β

j)2=2

3B2 (9.6)

and so the spatial isotropy manifests if and only if the sub-vector βjis zero. From this, in particular, we can see that the chr.inv.-tensorΩiΩ

k− 13 h

ki ΩjΩ

j becomes zero only with the chr.inv.-vector Ωj .

It is possible to say that the chr.inv.-invariants Π=D ljD

jl −

13D

2

and ΩjΩj , the chr.inv.-vector Ωi, and also the chr.inv.-tensors

Dki −

13 h

kiD and ΩiΩ

k− 13 h

ki ΩjΩ

j characterize the mechanical an-isotropy of the volume element (its kinematic anisotropy and itsdynamic anisotropy), while the chr.inv.-tensor Zki −

13 h

kiZ charac-

terizes its geometrical anisotropy. We shall consider the chr.inv.-vector Ωi first.

§4.10 Absolute rotations

In §3.9 we saw that in relativistic equations of dynamics Ωi plays apart akin to momentary angular velocity of the absolute rotationsof the reference frame in classic equations of dynamics. For thisreason we can call the chr.inv.-vectors Ωi and Ωk the chr.inv.-vectors of the angular velocities of the dynamic absolute rotation.


It is evident that they are the angular velocities, which can befound from experiments in mechanics. If we consider instead of theMetagalaxy, the regular test-bodies in an Earth-based laboratory,then the velocity can be found from a Foucault pendulum experi-ment, for instance. It is known (for instance, see [7], p. 183), thatthis velocity is different from the angular velocity of the kinematic“absolute” rotation by a quantity, which is, generally speaking,a function of a point. It is evident that the angular velocity ofthe kinematic absolute rotation can be characterized (to within achr.inv.-vector, the chr.inv.-derivative of which disappears) by thechr.inv.-vector ∗ω in the left side of the equation Rk(∗ω)= ∗rk(Ω)(7.18). Therefore it is possible to say that the equation (7.18) linksthe dynamic absolute rotations to the kinematic absolute rotations∗.In particular, this equation implies that the difference between thechr.inv.-vectors of the angular velocities of the dynamic absoluterotation and the kinematic absolute rotation is an irrotational chr.inv.-vector.

Now let us consider the angular velocity of the dynamic absoluterotation. We will use the condition Fi=0 (7.14). Throughout thefour-dimensional volume, where this condition is true, we have:

1. There isAik =

1

2

(∂vk∂xi

−∂vi∂xk

)

; (10.1)

2. Choice of the time coordinate in such way as to make equality(8.2) true throughout the said volume. So, because of (8.2),(8.8), and (10.1), we obtain

∂Aik∂t

= 0 (10.2)

and hence∗∂Aik∂t

= 0 . (10.3)

The latter equality and also (10.2), equivalent to it, because oftheir chr.inv.-tensor nature, are true for any choice of the timecoordinate.

∗We can see more clearly the geometrical sense of equation (7.18), whichlinks the dynamic absolute rotations to the kinematic absolute rotations, fromits “detailed” form ∗∇j (hkjD−Dkj)= ∗∇jAkj (7.19). This detailed form can beobtained after substituting Ri(∗ω)= ∗∇j(hijD−Dij) and ∗rk(Ω)= ∗∇jAkj into(7.18). Zelmanov uses Rk(∗ω) here as an alternative to denote the chr.inv.-rotor∗rk(∗ω) of the chr.inv.-vector ∗ω (see formula 17.15 of §2.17 for the details). —Editor’s comment. D. R.

4.10 Absolute rotations 197

Because of (9.1) of §3.9, we have

Ωi =1

2εijkAjk . (10.4)

Becauseεijk = 0 , (10.5)

or, in other word,

εijk = ±1√h, (10.6)

we obtain∗∂εijk

∂t= 0 , (10.7)


∗∂εijk

∂t= ∓

1

h

∗∂√h

∂t= −D

(

±1√h

)

. (10.8)

Hence, in general, we have

∗∂εijk

∂t= −εijkD . (10.9)

Therefore, formula (10.4), taking (10.3) into account, gives

∗∂Ωi

∂t= −ΩiD . (10.10)


∗∂

∂t

(√hΩi

)= Ωi

∗∂√h

∂t+√h∗∂Ωi

∂t=√hΩiD −

√hΩiD . (10.11)

So, we arrive at∗∂

∂t

(√hΩi

)= 0 , (10.12)

and, because of (13.2) of §3.13,

∗∂

∂t

(V Ωi

)= 0 , (10.13)

where V is the volume of the element of the accompanying space.Formula (10.13) implies that, in every element of the accompanyingspace, the chr.inv.-vector Ωi retains its direction and the non-zerocomponent of the vector is inversely proportionally to the volumeof the element.


Because of (10.12), we can write

Ωi =ξi√h, ξi6 ‖ t . (10.14)

Because of (10.10), the square of the chr.inv.-vector Ωi is∗∂

∂t

(hjlΩ

jΩl)= 2

(DjlΩ

jΩl −DhjlΩjΩl). (10.15)

Let us take coordinate axes at the given point in such a waythat at a given moment of time we have

Ω2 = Ω3 = 0 . (10.16)

Because of (10.10), this equality will hold for all moments oftime. Hence, at this point, we have

DjlΩjΩl = h11

(Ω1)2, (10.17)

∗∂

∂t

[h11(Ω1)2]=2[D11(Ω1)2−Dh11

(Ω1)2]=2

(D11h11

−D

)

h11(Ω1)2. (10.18)

We direct the axes x2 and x3 in those directions so that, at themoment of time we are considering, the axes are orthogonal to x1

at this point.Then, at that moment of time, we have

hik =

h11 0 0

0 h22 h230 h32 h33

(10.19)

and henceD11 =

(h11)2D11, (10.20)

h11 =1

h11, (10.21)

so thatD11h11

=D11

h11. (10.22)

We can re-write (10.18) in the form

1√h11(Ω1)2

∗∂√h11(Ω1)2

∂t= −

(

D−D11

h11

)

(10.23)

and with §2.12 as a basis, interpret it as follows:

4.11 Mechanical anisotropy and geometrical anisotropy 199

At any point at any moment of time the velocity of relativechanges of the ch.inv.-vector of the angular velocities Ωk

(or Ωi) of the absolute dynamic rotation is equal in its modulusand converse in its sign to the velocity of relative changes ofthe square of the surface element, which is orthogonal to thedirection of the vector Ωk.

The equality (10.23) is an analogue of the known theorem of ClassicMechanics, according to which the strength of a vortex remainsunchanged in time (for instance, see [58], p. 187).

§4.11 Mechanical anisotropy and geometrical anisotropy

In §4.10, we saw that the cosmological equation (7.14) providesa means to select one of the anisotropy factors for its studying,namely — the dynamic absolute rotation. At the same time, it isimpossible to divorce the deformation anisotropy and the curvatureanisotropy. This is true, in particular, because it is impossible todivorce quantities Ski and DDk

i (see §3.24). Their relation to oneanother, and also their relation to the dynamic absolute rotation,give the cosmological equation (7.17). We shall now consider itsconsequences.

A. We assume that at a moment of time at a given point we have

Dki −

1

3hkiD = 0 , (11.1)

then, generally speaking,∗∂

∂t

(Dki −

1

3hkiD

)6= 0 . (11.2)

So, in contrast to the dynamical absolute rotation, which cannot appear or disappear, the deformation anisotropy under thepresence of the absolute rotation or the curvature anisotropy(or, under both factors) can disappear for a moment and thenappear again.

B. If at the point we haveΩj = 0 , (11.3)

Zki −1

3hki Z ≡ 0 , (11.4)

then at this point,∗∂

∂t

(Dki −

1

3hkiD

)+D

(Dki −

1

3hkiD

)≡ 0 , (11.5)


or, because of (12.9) of §2.12,

∗∂

∂t

[

V(Dki −

1

3hkiD

)]

≡ 0 . (11.6)

Therefore, when the dynamical absolute rotation is absent andthe curvature anisotropy remains unchanged, the deformationanisotropy decreases with expansion of the volume element,and it increases with contraction of the volume.

C. If at this point condition (11.3) and

Dki −

1

3hkiD ≡ 0 (11.7)

are true, then (11.4) are true there as well. So, when thedynamical absolute rotation is absent and the deformation iso-tropy remains unchanged, the curvature isotropy manifests.In other words, mechanical isotropy leads to geometrical iso-tropy.

D. If at the point conditions (11.4) and (11.7) hold, then the con-dition (11.3) is true in the point. So, if the space deformationand the space curvature retain their isotropy unchanged, thenthe dynamic absolute rotation is absent.

§4.12 Isotropy and homogeneity

We suppose that conditions (11.3), (11.4), and (11.7) hold in a four-dimensional volume. Then we have

Π ≡ 0 , (12.1)

Akj ≡ 0 , ∗∇jAkj ≡ 0 , (12.2)

so the cosmological equations (7.15), (7.16), and (7.19) take theforms, respectively

∗∂D

∂t+1

3D2 = −

κ

2ρc2 + Λc2, (12.3)

1

3D2 + c2Z = κρc2 + Λc2, (12.4)

∗∂D

∂xi= 0 , (12.5)

4.12 Isotropy and homogeneity 201

while the cosmological equation (7.17) becomes an identity. Becauseof the first of the equalities (12.2), we can introduce the followingcoordinates

x0 = x0(x0, x1, x2, x3)

xi = xi , i = 1, 2, 3

(12.6)

so that throughout the aforementioned volume, we have

vi ≡ 0 , (12.7)

and soΣjk ≡ 0 . (12.8)

In this case, in accordance with §2.22, we have

Cki ≡ Zki (12.9)

and, because of (11.4),

Cki −1

3hki C ≡ 0 . (12.10)

Employing Shur’s theorem, we obtain

∂C

∂xi≡ 0 . (12.11)

Hence, going over to arbitrary coordinates and taking (12.9) intoaccount, throughout this volume we have

∗∂Z

∂xi= 0 . (12.12)

Formula (12.4), because of (12.5) and (12.2), leads to

∗∂ρ

∂xi= 0 . (12.13)

We have now obtained equations for a homogeneous model. Wecan transform the equations to the form (17.1) and (17.2) of §1.17,using the cosmic universal time coordinates (such time coordinatescan be introduced, because of 7.14 and 11.3). Then we obtain

∗∂

∂t=

∂

∂t,

∗∂

∂xi=

∂

∂xi(12.14)


and hence (see 7.13)

∂D

∂t+1

3D2 = −

κ

2ρc2 + Λc2, (12.15)

1

3D2 + c2C = κρc2 + Λc2, (12.16)

∂D

∂xi= 0 ,

∂C

∂xi= 0 ,

∂ρ

∂xi= 0 , (12.7)

∂ρ

∂t+Dρ = 0 . (12.18)

So, if the conditions (4.4–4.6), linked to our suppositions aboutthe properties of matter, are true, then the isotropy of the finitefour-dimensional volume implies that this volume is homogeneous.

§4.13 Stationarity in a finite volume

We assume that a substance throughout a finite four-dimensionalvolume, undergoes no deformations

Dik ≡ 0 , (13.1)

so, as it is evidently, we have

hik 6 ‖ t . (13.2)

Because of (7.13) and (10.10), we obtain, respectively

ρ 6 ‖ t , Ωk 6 ‖ t . (13.3)

Thus, this is a statistical case. The cosmological equations ofgravitation in this case can be written as follows

κ

2ρc2 = 2ΩjΩ

j + Λc2, (13.4)

3ΩjΩj + c2Z = κρc2 + Λc2, (13.5)

2(ΩiΩ

k −1

3hkiΩjΩ

j)+(Zki −

1

3hkiZ

)= 0 , (13.6)

∗rk(Ω)= 0 . (13.7)

We also assume that everywhere in the volume we are consider-

4.14 Transformations of the mean curvature of space 203

ing the conditionΩk = 0 (13.8)

is true. Then equation (13.7) becomes an identity, while equations(13.4), (13.5), and (13.6) take the forms, respectively

κ

2ρ = Λ , (13.9)

Z = κρ+ Λ , (13.10)

Zki −1

3hkiZ = 0 . (13.11)

Because of (13.8), we can introduce coordinates (12.6), wherethe conditions (12.7) are true. Then the conditions (12.8) and (12.9)will be true as well, because of (13.11), the equalities (12.10–12.13)∗.Using cosmic universal time, we can express Z and Zki in (13.10)and (13.11) through C and Cki . Inspecting equations (7.1) and (7.2)of §1.7 and comparing them with the foregoing, we see that the vol-ume we are considering can be interpreted as the Einstein model.

As a result, if the conditions (4.4–4.6), linked to suppositionsabout properties of matter, are true, and the absolute dynamicalrotation (13.8) is absent, then the condition of stability leads to theEinstein model. So, if non-empty static models, different from theEinstein model, can exist under the conditions (4.4–4.6), then theyare models with dynamic absolute rotation.

§4.14 Transformations of the mean curvature of space

Before we consider transformations of the volume of any spaceelement in time, we are going to deduce numerous consequences ofthe cosmological equations (7.13), (7.15), (7.16), here and in §4.15.Eliminating the cosmological constant from the last two equations,we can write

−∗∂D

∂t+ c2Z −

1

2Π + 3ΩjΩ

j = Π− 2ΩjΩj +

3κ

2ρc2. (14.1)

Differentiating (7.16) term-by-term, we obtain

2

3D∗∂D

∂t+

∗∂

∂t

(c2Z −

1

2Π + 3ΩjΩ

j)= κc2

∗∂ρ

∂t. (14.2)

As a result, multiplying (14.1) by 23 D term-by-term and sum-

∗The last two of the equalities are direct consequences of (13.9) and (13.10).


ming with (14.2), after taking (7.13) into account, we obtain( ∗∂

∂t+2

3D

)(c2Z −

1

2Π + 3ΩjΩ

j)=2

3D(Π− 2ΩjΩ

j). (14.3)

If, at a given point, we have

Π− 2ΩjΩj ≡ 0 , (14.4)

then it is evident that at this point we obtain( ∗∂

∂t+2

3D

)(c2Z −

1

2Π + 3ΩjΩ

j)= 0 , (14.5)

∗∂

∂t

[3√h(c2Z −

1

2Π + 3ΩjΩ

j)]

= 0 , (14.6)

or, after the primary time coordinate has been used (see §4.8),

∂

∂t

[3√h(c2C −

1

2Π + 3ΩjΩ

j)]

= 0 . (14.7)

In the case where (14.4) is a consequence of the equalities

Π ≡ 0 , ΩjΩj ≡ 0 , (14.8)

which imply that the space is isotropic at this point (see §3.11),equations (14.6) and (14.7) take the forms, respectively

∗∂

∂t

(3√hZ)= 0 , (14.9)

∂

∂t

(3√hC)= 0 . (14.10)

Thus we see that the equalities (14.5) and (14.6) are general-izations of the equalities (17.4) and (17.6) of §1.17. So (17.4) and(17.6) can be realized at any given point, if the space is isotropic atthis point and the isotropy remains unchanged.

§4.15 When the mass of an element and its energy remain un-changed

Formula (7.13) leads to∗∂

∂t

(√hρ)= 0 (15.1)

4.15 When the mass and energy remain unchanged 205

and, because of (12.7) and (12.8) of §2.12, we obtain∗∂M

∂t= 0 , (15.2)

where M is the mass of a volume element of the accompanyingspace, or equivalently, the mass of an element of the substance inthe space.

Because this space accompanies the substance (in other words,this is the accompanying space (see formula 6.2 of §3.6), we have

∗dM

dt=

∗∂M

∂t, (15.3)

so that∗dM

dt= 0 . (15.4)

Because the energy of the element is

E =Mc2, (15.5)we have

∗dE

dt= 0 . (15.6)

Instead of the equality (7.13), which is a particular case of (12.9)in §3.12, we can use the equality

( ∗dE

dt

)

fix

= 0 , (15.7)

which is a particular case of (13.18) in §3.13∗. Because we considerall in the accompanying space, we shall drop the suffix “fix”.

Formulae (15.4) and (15.6) lead to the equalities

dM

dt= 0 , (15.8)

dE

dt= 0 (15.9)

for any choice of the time coordinate, so the mass and energy ofthe element of the substance remain unchanged.

Finally, we can write the density of the element at this point,because of (13.2) and (13.3) of §3.13, in the form

ρ =μ√h, μ 6 ‖ t . (15.10)

∗Formula (13.18) is a consequence of (12.9) in §3.12 in the same Chapter 3.


§4.16 Criteria of the extreme volumes

Looking at formula (12.7) of §2.12 we can see that the value of thespace volume element is different from

√h by a multiplier, which is

independent of the time coordinate. For this reason, in studying theevolution of the value of a volume element, it would be sufficientto consider changes of

√h in time.

First, let us consider criteria for the extreme states∗ of thevolume of any given element during its transformations in time.Having chosen spatial coordinates so that the condition

√h 6≡ 0 (16.1)

is true at the point we are considering, we can write criteria forthe extrema as follows: the necessary criteria of the minimum

∂√h

∂t= 0 ,

∂2√h

∂t2> 0 , (16.2)

the sufficient criteria of the minimum

∂√h

∂t= 0 ,

∂2√h

∂t2> 0 , (16.3)

the necessary criteria of the maximum

∂√h

∂t= 0 ,

∂2√h

∂t26 0 , (16.4)

the sufficient criteria of the maximum

∂√h

∂t= 0 ,

∂2√h

∂t2< 0 . (16.5)

On the one hand, we have

∗∂√h

∂t=

c2

c2 − w∂√h

∂t,

∗∂2√h

∂t2=

(c2

c2 − w

)2(c2

c2 − w∂w

∂t

∂√h

∂t+∂2√h

∂t2

)

,

(16.6)

and on the other hand∗∂√h

∂t=√hD ,

∗∂2√h

∂t2=√h

(∗∂D∂t

+D2

)

, (16.7)

∗Here and in what follows, where we do not say the inverse, we mean regularextrema.

4.16 Criteria of the extreme volumes 207

so we obtain

∂√h

∂t=(1−

w

c2

)√hD ,

∂2√h

∂t2=√h

[(1−

w

c2

)2(∗∂D∂t

+D2

)

−D∂w

∂t

]

.

(16.8)

Limiting our tasks by the extreme states of the finite volume ofthe element (hence, its density is finite as well), we can see that theconditions (16.2–16.5) are equivalent to the conditions, respectively

D = 0 ,∗∂D

∂t> 0 , (16.9)

D = 0 ,∗∂D

∂t> 0 , (16.10)

D = 0 ,∗∂D

∂t6 0 , (16.11)

D = 0 ,∗∂D

∂t< 0 . (16.12)

The criteria, and hence the “maxima” and “minima” are inde-pendent of our choice of time coordinate. Note that the criterionfor non-accelerated transformations of the volume (this is the nec-essary criterion for the inflection point of the function

√h) is

∂2√h

∂t2= 0 (16.13)


∗∂D

∂t+D2 −

(c2

c2 − w

)2D∂w

∂t= 0 , (16.14)

so the accelerated and non-accelerated transformations are depen-dent on our choice of time coordinate. Taking a time coordinateso that

w = 0 , (16.15)

we transform (16.14) into the chr.inv.-condition∗∂D

∂t+D2 = 0 , (16.16)

where, as well as for the extrema, we suppose√h finite.


§4.17 The evolution of the volume of an element

Considering the cosmological equations, we see that the function√h of t is clearly linked to only ρ, Π, ΩjΩj , and Z by equations

(7.13), (7.15), and (7.16). In this section and we shall find thelimitations the equations impose on transformations of the volumeelement when the spatial coordinates remain unchanged and thetime coordinate changes arbitrarily. In this study, we consider theevolution not of the quantity

√h, but the quantity∗

η =6√h . (17.1)

Denoting chr.inv.-differentiation with respect to time by an ast-erisk, we can write

D =∗∂ ln

√h

∂t= 3

∗η

η, (17.2)

∗∂D

∂t= 3

( ∗∗η

η−

∗η2

η2

)

. (17.3)

It is not difficult to deduce, with√h (and, hence η) finite, that

the minimum or the maximum of one of the quantities leads to theminimum or the maximum of the other. So conditions (16.9–16.12)are equivalent to, respectively

∗η = 0 ,

∗∗η > 0 , (17.4)

∗η = 0 ,

∗∗η > 0 , (17.5)

∗η = 0 ,

∗∗η 6 0 , (17.6)

∗η = 0 ,

∗∗η < 0 . (17.7)

Let us introduce the auxiliary quantities τ and ξ (we will referto the latter as the criterion of the curvature) so that

Π− 2ΩjΩj =

3

2

τ

η, (17.8)

c2Z −1

2Π + 3ΩjΩ

j = 3ξ

η2. (17.9)

∗Compare it with R in the regular equations for homogeneous models.

4.17 The evolution of the volume of an element 209

Then equations (7.13), (7.15), (7.16) take the forms, respec-tively∗

∗ρ + 3

∗η

ηρ = 0 , (17.10)

3

∗∗η

c2η+

3

2c2τ

η= −

κ

2ρ+ Λ , (17.11)

3

∗η2

c2η2+ 3

ξ

c2η2= κρ+ Λ , (17.12)

and the equation (14.3), after its multiplication by η2 term-by-term, takes the form

∗ξ = τ

∗η . (17.13)

Any of the four equations (17.10–17.13) can be obtained as aconsequence of the other three. So equation (17.12), because of eq-uations (17.10) and (17.13), is the first integral of equation (17.11).Therefore we can, in particular, take (17.10), (17.12), (17.13) as theinitial equations. Instead of the latter, we can also take

ξ =

∫τ η dt , (17.14)

where the dot implies regular differentiation with respect to anarbitrary time coordinate. Eliminating the curvature criterion, wearrive at two cosmological equations, namely

∗ρ+3

∗η

ηρ = 0

3

∗η2

c2η2+

3

c2η2

∫τ η dt = κρ+ Λ

, (17.15)

which are analogous to (6.1) of §1.6. Next, eliminating ρ, we obtainone equation with respect to η (see 15.10)

3

∗η2

c2η2+

3

c2η2

∫τ η dt =

κμ

η3+ Λ , (17.16)

which leads to

ρ =μ

η3, μ 6 ‖ t . (17.17)

∗Compare the equations with equations (4.2), (5.1), (5.4) for homogeneousmodels (see §1.4 and §1.5).


To solve this equation we need to know τ as a function of t orη. At the same time, presupposing something about the quantitiesτ and ξ, we can obtain something about the evolution of η. Thiswill be one of our tasks. Moreover, in that study we will supposethat η does not remain fixed during the finite time interval we areconsidering.

Let us split the interval of changes of the time coordinate t(which can be finite or infinite), where the essential positive fun-ction

η = η (t) (17.18)

is defined, into the minimum number of time intervals where thisfunction is monotonic. Then at one of the boundaries of each the in-tervals (the intervals can be finite or infinite) we have the minimumnumerical value of the function, and at the other boundary, itsmaximum numerical value. We suppose this function continuouseverywhere and its derivative continuous under any non-zero nu-merical value of the function (under any finite numerical valueof the density, in other words). Then 6 kinds of the minimumnumerical values of the function η= η (t) can result. We label thekinds a, m, p, q, r, s. The maximum numerical values can be of 3kinds, and we call them A, D, M . In this terminology, we specify:

a — the non-zero asymptotic numerical value which thefunction approaches from above (under time coordinatechanges in its positive or negative direction);

m — the non-zero minimum of the function;p — its zero asymptotic value;q — zero value of the function when its derivative is zero;

r — zero value of the function when its derivative is non-zeroand finite;

s — zero value of the function when its derivative becomesinfinite;

A — the finite asymptotic value the function ap-proaches from below (under time coordinate changes inits positive or negative direction);

D — the infinite value of the function;

M — the finite maximum of the function.

In accordance with this terminology, we will refer to states ofthe volume element as follows. States of finite density:

4.18 Changes of the criterion of the curvature 211

D — the limit state of the infinite rarefaction;

M — the state of the maximum volume;

A — a state, where the volume remains unchanged;

a — a state, where the volume remains unchanged;

m — the state of the minimum volume.

States of infinite density (p, q, r, and s):

p — the asymptotic state of infinite density;q — the minimum state of infinite density;

r — the collapsed state of infinite density;

s — the special state of infinite density;

All the states can appear in the theory of non-empty homoge-neous models (see §1.7) except the states p, q, and r.

§4.18 Changes of the criterion of the curvature

We fix the spatial coordinates, so we consider xi, τ , and η as func-tions of only the time coordinate. At the same time, the time coord-inate is linked to the function η by a mutually-unique relation inevery interval, where this function undergoes monotonic changes.Therefore, in each of the aforementioned intervals, we can considerξ and τ as functions of η. Inspecting (17.13), we see that

∂ξ

∂η= τ , ξ =

∫τdη . (18.1)

We use the equations

3

c2∗∗η +

3

2c2τ = −

κ

2

μ

η2+ Λη , (18.2)

3

c2∗η2 +

3

c2ξ = κ

μ

η+ Λη2, (18.3)

which have the same power (for finite numerical values of η) asthose equations which can be deduced by eliminating ρ from(17.10–17.12). Because of (18.1), equation (18.3) is the first integralof equation (18.2), so we could limit our tasks by considering onlyequation (18.3). However some consequences could be obtainedmore easily from (18.2), so we also consider this equation.

The previous section described some limitations on the functionη and its derivative. Taking the limitations, and because of (17.12),


(18.3), and since w<c2 is continuous, we deduce that, for η 6=0, thequantity ξ is a continuous function of t. As a result, the curvaturecriterion is a continuous function of η in any finite interval of η,where a zero-point is absent. So we will consider those intervalswhere η is a monotonic function of t. Let us consider different casesof this function.

Case I:τ ≡ 0 . (18.4)

This case manifests, in particular, under the conditions

Π ≡ 0 , ΩjΩj ≡ 0 , (18.5)

i. e. when the mechanical isotropy, and hence the geometrical isot-ropy, remain unchanged at the given point we are considering(see §4.11). Because of (18.4), and taking (18.1) into account, weobtain

ξ = const T 0 . (18.6)

So, in this case, we have τ ≡ 0. In other words, three kinds ofthe curvature criterion are possible under conditions (18.5):

(1) the curvature criterion is positive;

(2) the curvature criterion equals zero;

(3) the curvature criterion is negative.

We introduce notation for the cases: I1, I2, I3. In addition, weintroduce

R = R (t) (18.7)

so that, at the point we are considering, we have∗η

η=

∗R

R,

ξ

c2η2=

k

R2, k = 0;±1 . (18.8)

Besides these, choosing time coordinates so that

w ≡ 0 (18.9)

at this point, we bring the cosmological equations (7.10–7.12) forthis volume element into their known form in the theory of a ho-mogeneous universe.

Case II:τ > 0 . (18.10)

4.19 The states of the infinite density 213

This case manifests, in particular, under the conditions

Π 6≡ 0 , ΩjΩj = 0 , (18.11)

i. e. under anisotropic (generally speaking) deformations in the ab-sence of the dynamical absolute rotation at the point we are con-sidering.

Because of (18.10), and taking (18.1) into account, we concludethat the curvature criterion does not decrease if η increases, andalso that the curvature criterion does not increase if η decreases. Itis evident that the following cases are possible:

(1) the curvature criterion is positive for the minimum numer-ical value of η, so the curvature criterion is positive for allnumerical values of η in the interval we are considering;

(2) the curvature criterion equals zero at one of the boundaries,or inside the interval. In particular, when η transits fromsmaller numerical values to the larger ones, the negative cur-vature criterion can become positive;

(3) the curvature criterion is negative for the maximum numeri-cal value of η, and hence the curvature criterion is also neg-ative for all numerical values of η in the interval.

We will refer to the cases as II1, II2, and II3.

Case III (the general case):

τ T 0 . (18.12)

We also have three different cases here. So we will refer to thecases as III1, III2, III3:

(1) the curvature criterion is positive throughout the interval ofη, the boundaries included;

(2) the curvature criterion takes numerical values, which can bezero. In particular, the curvature criterion can change its sign;

(3) the curvature criterion remains negative throughout the in-terval of numerical values of η, the boundaries included.

§4.19 The states of the infinite density

When a volume element we are considering approaches the asymp-totic state of infinite density, we have

η → ηp = 0 ,∗η →

∗ηp = 0 ,

∗∗η →

∗∗η p = 0 . (19.1)


When the volume element approaches the minimum state ofinfinite density, we have

η → ηq = 0 ,∗η →

∗ηq = 0 ,

∗∗η →

∗∗η q > 0 . (19.2)

In both cases, because of (18.2) and (18.3), we obtain

τ → −∞ , (19.3)

ξ → +∞ . (19.4)

It is easy to see that the formula (19.4) leads to (19.3), becausethe curvature criterion is finite and continuous when η 6=0.

When the element approaches the collapsed state of infinitedensity, we have

η → ηr = 0 ,∗η →

∗ηr 6= 0 (19.5)

and (18.3) leads to (19.4) and, hence (19.3). From the foregoing weconclude that: in the cases I1, I2, I3, II1, II2, II3, III3 the asymptotic,minimum, and collapsed states of the infinite density are impos-sible; in the cases III1 and III2 the aforementioned states are pos-sible only under the specific evolution of the curvature criterion.On the possibility of states which are not prohibited by the fore-going results, see §4.25.

When the volume element approaches the special state of infi-nite density, we have

η → 0 ,∗η → ±∞ . (19.6)

It is evident that under the specific sequence by which∗η ap-

proaches infinity, the special states of infinite density are conceiv-able in all the cases of §4.18 (see §4.25 for the details).

§4.20 The ultimate states of the infinite rarefaction

We assume thatη →∞ . (20.1)

Let us consider those cases where the cosmological constant ispositive, zero, and negative.

If the cosmological constant is positive, then (18.2) and (18.3)under the condition (20.1) give, respectively

(∗∗η +

1

2τ)→ +∞ , (20.2)

4.21 An unchanged volume and the minimum volume 215

(∗η2 + ξ

)→ +∞ . (20.2)

Hence, under the specific changes of∗η and

∗∗η , the element can

approach the ultimate state of infinite rarefaction in all 9 cases wehave considered in §4.18 (see also §4.25 and §4.26).

If the cosmological constant equals zero, then (18.2) and (18.3),under the condition (20.1), give

(∗∗η +

1

2τ)→ 0 , (20.4)

(∗η2 + ξ

)→ 0 . (20.5)

So the curvature criterion approaches a nonpositive numericalvalue, while the derivative of the curvature criterion, depending on

the changes of∗∗η , can evolve in different ways. Hence, the element

can not approach the ultimate state of infinite rarefaction in thecases I1, II1, III1.

Finally, if the cosmological constant is negative, then (18.2) and(18.3), under the condition (20.1), respectively lead to

(∗∗η +

1

2τ)→ −∞ , (20.6)

(∗η2 + ξ

)→ −∞ . (20.7)

From (20.7) we obtain

ξ → −∞ , (20.8)

so τ cannot remain nonnegative. So the element cannot approachthe ultimate state of infinite rarefaction in the cases I1, I2, I3, II1,II2, II3, III1.

§4.21 The states of unchanged volume and of the minimum vol-ume

When a volume element approaches a state of unchanged volume,we have

∗η → 0 ,

∗∗η → 0 , (21.1)

so formulae (18.2) and (18.3) give, in their limits,

3

2c2τ = −

κμ

η2+ Λη , (21.2)


3

c2ξ =

κμ

η+ Λη2. (21.3)

In the state of minimum volume, we have∗η = 0 ,

∗∗η > 0 (21.4)

and hence (18.2) transforms into

3

2c2τ 6 −

κμ

η2+ Λη , (21.5)

while (18.3) transforms into (21.3). So, both in an unchanged vol-ume and in minimum volume we have

3

2c2τ < Λη , (21.6)

3

c2ξ > Λη2. (21.7)

Let us consider the cases where the cosmological constant ispositive, zero, or negative, just as we did in §4.20.

With the cosmological constant positive, equation (21.7) impliesthat the transit through the state of minimum volume and also theasymptotic approach to the state of unchanged volume from aboveare impossible in the cases I2, I3, II2, II3, III3, while the asymptoticapproach from below is impossible in the cases I2, I3, II3, III3.

If the cosmological constant is zero, equations (21.6) and (21.7)imply that the transit through the state of minimum volume andalso the asymptotic approach to the state of unchanged volume(from above or below) are impossible in the cases I1, I2, I3, II1, II2,II3, III3.

Finally, if the cosmological constant is negative, equations (21.6)and (21.7) imply that the transit through the state of the minimumvolume and also the asymptotic approach to the state of unchangedvolume are impossible in the cases I1, I2, I3, II1, II2, II3.

§4.22 The states of the maximum volume

In the state of maximal volume, we have∗η = 0 ,

∗∗η 6 0 (22.1)

so (18.2) and (18.3) give3

2c2τ > −

κμ

η2+ Λη , (22.2)

4.23 The transformations, limited from above and below 217

3

c2ξ =

κμ

η+ Λη2, (22.3)

so that3

c2τ TΛη , (22.4)

3

c2ξ > Λη2. (22.5)

The result (22.5) implies that, with the cosmological constantnonpositive, the state of the maximum volume is impossible in thecases I2, I3, II3, III3. If the cosmological constant is strictly neg-ative, then equations (18.2) and (18.3) permit the state of maximumvolume in all 9 cases we have considered in §4.18 (see also §4.25and §4.26).

§4.23 The transformations, limited from above and below

We will now consider cases where monotonic changes of η arelimited from above and below.

The lower boundary is the finite asymptotic or the finite min-imum numerical value η1, while the upper boundary is the finiteasymptotic or the maximum numerical value η2. Then it is evi-dent that

η1 < η2 , (23.1)

∗η1 = 0 =

∗η2 , (23.2)

∗∗η1 > 0 >

∗∗η2 . (23.3)

If the cosmological constant is nonnegative, then equation(18.2) gives

τ1 < τ2 . (23.4)

If the cosmological constant is nonpositive, then equation(18.3) gives

ξ1 > ξ2 . (23.5)

Because of (23.4) and (23.5), the kinds of changes of η we areconsidering are impossible in the cases I1, I2, I3 for a strictly positivecosmological constant, and in the cases I1, I2, I3, II1, II2, II3 for anonpositive cosmological constant.


§4.24 The area, where deformations of an element are real

In §4.19–§4.23 we ascertained numerous prohibitions and limitat-ions on the evolution of the volume element, due to equations (18.1)and (18.3) or equivalently, by equations (7.13), (7.15), (7.16). Sothe problem we have stated in §4.17 has been solved. It is easyto see the prohibitions are generalizations of those prohibitionson the evolution of any element of a homogeneous universe, thecosmological equations resulted when the pressure becomes zero.

The prohibitions can also be found by considering an area ofreal deformations of the element in the plane η, ξ, the area, where

η > 0 , (24.1)

∗η2 > 0 . (24.2)

Because of (18.3), equation (24.2) leads to

ξ 6c2

3

(κμ

η+ Λη2

)

. (24.3)

So the area of real deformations is bounded by the ordinate axisand the ultimate curve

ξ =c2

3

(κμ

η+ Λη2

)

, η > 0 . (24.4)

Let us consider this curve in detail. If the cosmological constantis positive, then the whole curve is located in the first quadrant,is convex everywhere with respect to abscissa axis, and it has aminimum when

η = 3

√κμ

2Λ. (24.5)

If η approaches zero (the ordinate axis is the asymptote), thenwe have

ξ → +∞ ,∂ξ

∂η→ −∞ , (24.6)

and if η approaches infinity, then

ξ → +∞ ,∂ξ

∂η→ +∞ . (24.7)

If the cosmological constant is zero, then the whole ultimatecurve lies in the first quadrant and is convex with respect to the

4.24 The area, where deformations of an element are real 219

abscissa axis. The curve is a branch of an equilateral hyperbola,the asymptotes of which are the coordinate axes, so that: (1) ifη approaches zero, then conditions (24.6) are true, and (2) if ηapproaches infinity, then

ξ → 0 ,∂ξ

∂η→ 0 . (24.8)

Finally, if the cosmological constant is negative, then the ultima-te curve is monotonic (like the previous case) decreasing function.In this case the curve is convex with respect to the abscissa axisand intersects the axis at the point

η = − 3

√κμ

Λ, (24.9)

which is, hence, the inflection point. If η approaches zero, we have,just as in the previous cases, the conditions (24.6), so the ordinateaxis is the asymptote. If η approaches infinity, then

ξ → −∞ ,∂ξ

∂η→ −∞ . (24.10)

In the case of the positive cosmological constant, the real statesof infinite rarefaction are points at infinity on the straight line

η = +∞ . (24.11)

In the case where the cosmological constant is zero, the statesare points on the straight half-line

η = +∞ , ξ 6 0 . (24.12)

Finally, in the case of the negative cosmological constant, thestates are the points

η = +∞ , ξ = −∞ . (24.13)

The special states of infinite density are all points on the ordina-te axis. The asymptotic, minimum, and collapsed states of infinitedensity are the points

η = 0 , ξ = +∞ (24.14)

on this axis.The states of unchanged volume, the minimum volume, and the

maximum volume are points on the ultimate curve. The asymptotic


approach to a state of unchanged volume is, it is easy to under-stand, the approach to the point of this state along a curve, a tan-gential line to which at this point is the same as the tangential lineto the ultimate curve.

§4.25 The kinds of monotonic transformations of a volume

We will label each kind of evolution of the volume element by var-ious letters, denoting the states the element evolves through (thestates are boundaries of the monotonic change of the volume). Inthis terminology, the kinds of evolution we consider in the theoryof a non-empty homogeneous universe will be denoted as follows:

A1 — sA (expansion) or As (contraction);

A2 — aD (expansion) or Da (contraction);

M1 — sD (expansion) or Ds (contraction);

M2 — DmD ;

O1 — sMs ;

O2 — . . .mMmM. . .

It is evident that the number of kinds of evolution of the elementwhich are conceivable inside the interval of its monotonic transfor-mations is 18, if the volume expands, and the same number if thevolume contracts (6 kinds of minimum value and 3 kinds of themaximum value). Let us make a list of the kinds of evolution whenthe element expands (by changing the subscript letters we obtainthe corresponding kinds of contraction).

1. Transformations of the volume, limited by its finite value onlyfrom below: aD, mD.

2. Transformations of the volume, limited by its finite valuefrom below and above: aA, aM , mA, mM .

3. Transformations of the volume, limited by its finite value onlyfrom above: pA, pM , qA, qM , rA, rM , sA, sM .

4. Unlimited transformations: pD, qD, rD, sD.

The kinds aD, mD and the corresponding kinds of contractionare impossible in the cases (see §4.20 and §4.21)

Λ > 0 : I2, I3; II2, II3; III3

Λ = 0 : I1, I2, I3; II1, II2, II3; III1, III3

Λ < 0 : I1, I2, I3; II1, II2, II3; III1,

. (25.1)

4.25 The kinds of monotonic transformations of a volume 221

The kinds aA, aM , mA, mM and the corresponding kinds ofcontraction are impossible in the cases (see §4.21, §4.22, §4.23)

Λ > 0 : I1, I2, I3; II2, II3; III3

Λ = 0 : I1, I2, I3; II1, II2, II3; III3

Λ < 0 : I1, I2, I3; II1, II2, II3

. (25.2)

The kinds pA, pM , qA, qM , rA, rM and the corresponding kindsof contraction are impossible in the cases (see §4.19, §4.21, §4.22)

Λ T 0 : I1, I2, I3; II1, II2, II3; III3 . (25.3)

The kind sA and the corresponding kind of contraction areimpossible in the cases (see §4.19 and §4.21)

Λ > 0 : I2, I3; II3; III3

Λ = 0 : I1, I2, I3; II1, II2, II3; III3

Λ < 0 : I1, I2, I3; II1, II2, II3

. (25.4)

The kind sM and the corresponding kind of contraction areimpossible in the cases (see §4.19 and §4.22)

Λ > 0 : I2, I3; II3; III3 . (25.5)

The kinds pD, qD, rD and the corresponding kinds of contrac-tion are impossible in the cases (see §4.19 and §4.20)

Λ > 0 : I1, I2, I3; II1, II2, II3; III3

Λ 6 0 : I1, I2, I3; II1, II2, II3; III1, III3

}

. (25.6)

The kind sD and the corresponding kind of contraction areimpossible in the cases (see §4.19 and §4.20)

Λ = 0 : I1; II1; III1

Λ < 0 : I1, I2, I3; II1, II2, II3; III1

}

. (25.7)

We have listed those cases for each type of evolution prohibitedby the limitations we have obtained from equations (7.13), (7.15),(7.16) in §4.19–§4.23. Considering the ultimate curve in an areaof real deformations (see §4.24), we see that each kind is possible(more exactly, permitted by equations 7.13, 7.15, and 7.16) in all


cases where the type is not prohibited by the aforementioned limit-ations.

We give two tables in which each case is specified with the typeof evolution (for brevity, we include only the kinds of expansion),which are permitted by equations (7.13), (7.15), (7.16) or equiva-lently, by equations (18.1) and (18.3). The meanings of the under-lined and the bracketed terms are given in §4.27.

§4.26 The possibility of an arbitrary evolution of an element ofvolume

We have considered the limitations that equations (7.13), (7.15),(7.16) place upon the evolution of η, if the time coordinate changeswhen the spatial coordinates are fixed. Let us recall those consider-ations, whereby we conclude that the other cosmological equa-tions do not have additional limitations on the evolution of η atthe given point.

It is known [59] that we can always∗ introduce coordinates(x0, x1, x2, x3), where the conditions

∂(gμν√−g

)

∂xν= 0 (26.1)

hold (the harmonic coordinates). The equations of gravitation inthese coordinates can be written in the form

1

2˜ gμν − Γμαβ Γ

νεζ g

αεgβζ + Λ gμν = κ(Tμν −

1

2gμν T

), (26.2)

where we denote

≡ gαβ∂2

∂xα∂xβ, (26.3)

and hence the equations of gravitation can be transformed intotheir regular form

∂2gμν

∂x3∂x3= Fμν

(

x0, x1, x2, x3; g00, g01, . . . , g33;

∂g00

∂x0,∂g00

∂x1, . . . ,

∂g33

∂x3;

∂2gμν

∂x0∂x0,∂2gμν

∂x0∂x1, . . . ,

∂2gμν

∂x3∂x1

)

.

(26.4)

∗However, generally speaking, it is not possible in every reference frame.

4.26 An arbitrary evolution of an element of volume 223

I1 I2 I3

aD (A2), mD [M2]

Λ > 0 sA (A1)

sM [ O1]sD (M1) sD (M1) sD (M1)

Λ = 0sM [ O1]

sD (M1) sD (M1)

Λ < 0sM [ O1] sM [ O1] sM [ O1]

Table 4.1 Kinds of evolution of the volume element in Case I, i. e.for fixed mechanical isotropy and fixed geometrical isotropy.

II1 II2 II3

aD (A2), mDaA,aM ,mA,mM

Λ > 0 sA (A1) sA (A1)

sM sM [ O1]sD (M1) sD (M1) sD (M1)

Λ = 0

sM [ O1] sM [ O1]sD (M1) sD (M1)

Λ < 0

sM [ O1] sM [ O1] sM [ O1]

Table 4.2 Kinds of evolution of the volume element in Case II, i. e.for anisotropic deformations in the absence of dynamical absoluterotation.


III 1

III 2

III 3

aD(A

2),mD

aD(A

2),mD

aA

,aM

,mA

,mM

aA

,aM

,mA

,mM

Λ>0

pA

,pM

,qA

,qM

,rA

,rM

pA

,pM

,qA

,qM

,rA

,rM

sA(A

1)

sA(A

1)

sMsM

pD

,qD

,rD

pD

,qD

,rD

sD(M

1)

sD(M

1)

sD(M

1)

aD(A

2),mD

aA

,aM

,mA

,mM

aA

,aM

,mA

,mM

Λ=0

pA

,pM

,qA

,qM

,rA

,rM

pA

,pM

,qA

,qM

,rA

,rM

sA(A

1)

sA(A

1)

sMsM pD

,qD

,rD

sD(M

1)

sD(M

1)

aD(A

2),mD

aD(A

2),mD

aA

,aM

,mA

,mM

aA

,aM

,mA

,mM

aA

,aM

,mA

,mM

Λ<0

pA

,pM

,qA

,qM

,rA

,rM

pA

,pM

,qA

,qM

,rA

,rM

sA(A

1)

sA(A

1)

sA(A

1)

sMsM

sMpD

,qD

,rD

sD(M

1)

sD(M

1)

Tab

le4.

3K

inds

ofev

oluti

onof

the

elem

ent

inC

aseIII

(the

gener

alca

se).

4.26 An arbitrary evolution of an element of volume 225

In such a case∗ we can use the general theorem concerning in-tegrals, which satisfy the given initial conditions for

x3 = a3 = const , (26.5)

gμν = fμν(x0, x1, x2) ,∂gμν

∂x3= f

μν3 (x0, x1, x2) . (26.6)

All 20 functions on the right sides of equalities (26.6) mustsatisfy† only 4 of the harmonic conditions (26.1). Thus, 16 of thefunctions and, hence, no less than 6 of the 0 functions fμν can begiven by our arbitrary choice.

Let us link the accompanying coordinate frame (x0, x1, x2, x3)we are considering to a harmonic coordinate frame by the generaltransformations

x0 = x0 (x0, x1, x2, x3)

xi = xi (x0, x1, x2, x3)

}

, (26.7)

and we choose coordinates in the harmonic coordinate frame and apoint in the accompanying coordinate frame

xi = ai = consti, (26.8)

so that this point would always be in the surface (26.5). Then atthis point, we have

gαβ = fαβ(x0) , (26.9)

where we denote‡

fαβ =

(

fμν∂xα

∂xμ∂xβ

∂xν

)

a

. (26.10)

Thus, 10 values of fαβ are linear functions of 10 values of(fμν)a,

6 of which can be given arbitrarily. For this reason, we can preas-sign the 6 values of fαβ instead of

(fμν)a.

The foregoing implies that at a given point of the space, preas-signing numerical values of w, v1, v2, v3 as functions of the time

∗We suppose that all the necessary requirements of continuity and differenti-ability are realized.

†Beside the regular requirements of continuity and that their continuous firstderivatives must exist.

‡The index a here implies that we consider all the functions, which havebeen contained by the brackets, along the world-line of the point (26.8) of theaccompanying space in the harmonic coordinate frame.


coordinate, we can also preassign no less than two values of hik orhik as functions of the time coordinate at this point. Alternatively,we can preassign two functions of the hik or hik, where we canchoose for one of the functions the value of η. In this case we have:

• A meaning for the limitations the cosmological equations(7.13), (7.15), (7.16) place upon the evolution of the numericalvalue of η — setting of a relation between the evolution of ηand the evolution of the curvature criterion;

• The other cosmological equations cannot give additional rela-tions between η and ξ, because in the opposite case the quan-tities would be defined by no less than two independent equa-tions, so it would be impossible to preassign η (t) arbitrarily;

• When η, during its evolution, transits from one interval of itsmonotonic transformations into another, any type of the mo-notonic transformations can be replaced by any other, whichis permitted for the given cosmological constant and the pre-assigned evolution of the curvature criterion.

The foregoing implies that, generally speaking, it is impossibleto preassign η for all elements of a finite volume.

In conclusion, let us make a list of data, which can be obtainedfrom the cosmological equations we considered as the equations forthe given point.

Let us suppose that at the given point we preassign

w = 0 , (26.11)

∂w

∂xi= 0 , (26.12)

vi = 0 , (26.13)

numerical values of μ, ν1, ν2, ν3, and also 5 of 6 numerical valuesof hik as functions of the time coordinate (in other words, we know5 functions f ik for the given point). Then we have:

• Equation (7.14) will be satisfied, because of (26.11) and (26.13);

• Equations (7.13) and (7.15) establish the 6th value∗ of all nu-merical values of hik and ρ ;

• Equation (7.15) establishes the value of Z ;

∗So we see that to presuppose all 6 values of f ik when (26.11) has beenpresupposed, is prohibited by the cosmological equations for the given point.

4.27 The kinds of evolution of the element 227

• Equation (7.17), because of (10.14), determines 5 values of allnumerical values of Zki −

13h

kiZ ;

• Equation (7.19) links the evolution of the given element tothe evolution of the neighbouring elements, if the relationbetween values of νi and the spatial coordinates we use isknown.

§4.27 The kinds of evolution of the element

We are now going to consider the kinds of evolution of the volumeelement throughout the interval of changes of η, where its numeric-al value and the substance’s density remain finite. It is evident thatthe limiting states in this case can be all kinds, aside of m and M ,so only 10 of 18 kinds of the monotonic expansion (or contraction,respectively) are evolution types throughout the interval we areconsidering (the kinds are underlined in Table 4.1, Table 4.2, andTable 4.3). Examining the types, we see that 7 of the 10 kindsare impossible for a homogeneous universe (these are aA, pA, qA,rA, pD, qD, rD), while the other 3 kinds aA, sA, sD coincide,respectively, with kinds A2, A1, M1, which are possible for a homo-geneous universe (the kinds have been contained by parentheses inthe Tables).

It is conceivable that cases can exist where the volume elementcan transit from the states m or M into only one of the variousstates of infinite density, unchanged volume, or infinite rarefaction(as a result of the monotonic transformation of the volume). In suchcases, the kind of monotonic transformations of the volume elementdefines the kind of its evolution throughout the interval we have aninterest in. For instance, if the volume element can transit fromthe state m into only the state D, then the presence of the kindmD implies the presence of the kind DmD, which coincides withkind M2. If the element can transit from the state M into only thekind s, then the presence of the kind sM implies the presence ofthe kind sMs, which coincides with kind O1, as shown in Table 4.1and Table 4.2 by the terms in square brackets.

The foregoing considerations are sufficient for determining allthe kinds of evolution:

• In the cases I1, I2, I3, II2, II3 under Λ S 0;

• In the case II1 under Λ6 0;

• In the case III3 under Λ> 0.


To ascertain the kinds of evolution:

• In the case II1 under Λ> 0,

• In the cases III1 and III2 under Λ T 0,

• In the case III3 under Λ< 0

we need to use our findings on the possibility of combining thekinds of monotonic transformations of the volume (see §4.26). Thisenables us to conclude that:

• In the cases II1, Λ> 0 and III3, Λ< 0, the volume element cantransit from any of the states a, m, s into any of the states A,M , D and back;

• In the cases III1, Λ> 0 and III2, Λ T 0, the volume element cantransit from any of the states a, m, p, q, r, s into any of thestates A, M , D and back;

• In the case III1, Λ6 0, the volume element can transit fromany of the states a, m, p, q, r, s into any of the states A, Mand back.

Note that the kind . . .mMmMmM. . . (i. e. the kind O2) is pos-sible in all cases.

§4.28 The roles of absolute dynamic rotation and deforma-tion anisotropy

To clarify the effects of dynamic absolute rotation and deformationanisotropy, let us consider the following cases: (1) dynamic absoluterotation is absent when the deformations are isotropic; (2) dynamicabsolute rotation is absent when the deformations are anisotropic;(3) dynamic absolute rotation is present when the deformations areanisotropic.

The dynamic absolute rotation is absent when the deformationsare isotropic

We assume that the mechanical isotropy and, hence, the geo-metrical isotropy remain unchanged at the given point we are con-sidering. It easy to see from §4.11 that the necessary and sufficientconditions for this conservation can be written in the form

Π ≡ 0 , ΩjΩj = 0 . (28.1)

Then, on one hand, we have

τ ≡ 0 , ξ = const , (28.2)

4.28 The roles of absolute rotation and anisotropy 229

and on the other hand

c2Z = 3ξ

η2. (28.3)

In §4.14 we saw (it is also seen from formula 28.3) that, if theisotropy is at the point and remains unchanged, then the meancurvature at the point transforms in company with transformationsof the volume of the element just as for a homogeneous universe.Considering formula (28.2), we obtain (see Table 4.1, taking 28.3into account) that the kinds of evolution of the volume element,possible for a given cosmological constant and mean curvature (inthe sense of its sign or if it is zero), are the same as for a homoge-neous universe. Moreover, we saw in §4.18 that equations (7.13),(7.15), (7.16), under the conditions (28.2) at the given point, can betransformed into their regular form for a homogeneous universe.

The dynamic absolute rotation is absent when the deformationsare anisotropic

We assume that the dynamic absolute rotation at this point isagain absent, while the deformation anisotropy is present

Π > 0 , ΩjΩj = 0 . (28.4)

Then, as it is easy to see, we have

τ > 0 , (28.5)

c2Z > 3ξ

η2. (28.6)

The deformation anisotropy, generally speaking, complicates thefunction of the evolution of the mean curvature from the evolutionof the volume of the element (see §4.14). In particular, it makeschanging of the sign of the mean curvature possible. If the dynamicabsolute rotation is absent, then the deformation anisotropy resultsin new kinds of evolution of the element (see Table 4.2), whichare absent for a homogeneous universe. For a positive cosmologicalconstant and always for the positive curvature criterion∗, the limit-ed from above and below monotonic transformations of the volumeof the element (the kinds aA, aM , mA, mM) are possible. So thekind O2 is also possible under the conditions. Note also that forany numerical value of the cosmological constant and an always

∗Hence, because of (28.6), for the always positive mean curvature.


nonpositive mean curvature, only those kinds of evolution are pos-sible which are also possible for a homogeneous universe for thesame cosmological constant and the same nonpositive curvature.

The presence of the dynamic absolute rotation when the deforma-tions are anisotropic

We assume that the deformation anisotropy and the dynamicalabsolute rotation are present at the point we are considering. Letus consider the general case, where Π can be greater than, equalto, or less than 2ΩjΩj . Then it is evident that we have

τ T 0 , (28.7)

c2Z T 3ξ

η2. (28.8)

Because of (28.7) (see Table 4.3), the kinds aA, aM , mA, mMand consequently the kind O2 are possible not only for a positivecosmological constant, but also when the constant becomes zeroor negative. Moreover, in the case of a nonnegative cosmologicalconstant, the curvature criterion does not always remain positive.This limitation is not present in the case of a strictly negativecosmological constant∗.

If dynamic absolute rotation is present, then the new states p,q, r and the associated types pA, qA, rA, pM , qM , rM , pD, qD, rDwill be possible.

Such are the various consequences of deformation anisotropyand dynamic absolute rotation.

♦

∗Whatever the sign of the curvature criterion, condition (28.8) permits apositive, zero, or negative mean curvature.

Bibliography

1. Tolman R.C. Relativity, thermodynamics, and cosmology. ClarendonPress, Oxford, 1934.

2. Milne E.A. Relativity, gravitation, and world-structure. ClarendonPress, Oxford, 1935.

3. Robertson H.P. Proc. Nat. Acad. Sci. USA, 15, 822, 1929.4. Einstein A. Berlinere Berichte, 142, 19175. Friedmann A. Zeitschrift fur Physik, 10, 377, 1922.6. Einstein A. Berlinere Berichte, 235, 1931.7. Eddington A. S. The mathematical theory of relativity. 3rd exp. ed.,

GTTI, Moscow, 1934.8. Levi-Civita T. Das absolute Differentialkalkul. Springer, Berlin, 1928.9. de Sitter W. Kon. Ned. Acad. Amsterdam Proc., 19, 1217, 1917.

10. de Sitter W. Monthly Notices, 78, 3, 1917.11. Lemaıtre G. Journal of Mathematical Physics, 4, 188, 1925.12. Friedmann A. Zeitschrift fur Physik, 21, 326, 1924.13. Lanczos K. Physikalische Zeitschrift, 23, 539, 1922.14. Weyl H. Physikalische Zeitschrift, 24, 230, 1923.15. Lemaıtre G. Ann. Soc. Sci. Bruxelles, 47A, 49, 1927; Monthly Notices,

91, 483, 1931.16. Lemaıtre G. Bull. Astron. Inst. Netherlands, 5, 273, 1930.17. Heckmann O. Gottingene Nachrichten, 97, 1932.18. Robertson H.P. Review of Modern Physics, 5, 62, 1933.19. Eddington A. S. Monthly Notices, 90, 668, 1930.20. Tolman R.C. Proc. Nat. Acad. Sci. USA, 14, 268, 1928.21. Tolman R.C. Proc. Nat. Acad. Sci. USA, 14, 701, 1928.22. Tolman R.C. Physical Review, 37, 1639, 1931.23. Tolman R.C. Physical Review, 38, 797, 1931.24. Tolman R.C. Physical Review, 39, 320, 1932.25. Milne E.A. Quart. Journal of Math., 5, 64, 1934.26. McCrea W.H. and Milne E.A. Quart. Journal of Math., 5, 73, 1934.27. Eigenson M.S. Zeitschrift fur Astrophysik, 4, 224, 1932.28. Eddington A. S. Relativity theory of protons and electrons. Cambridge

Univ. Press, Cambridge, 1936.29. Einstein A. and de Sitter W. Proc. Nat. Ac. Sci. USA, 18, 213, 1932.

232 Bibliography

30. Tolman R.C. Physical Review, 38, 1758, 1931.31. Lanczos K. Zeitschrift fur Physik, 17, 168, 1923.32. Hubble E.P. Astrophysical Journal, 64, 321, 1926.33. Hubble E.P. Proc. Nat. Acad. Sci. USA, 15, 168, 1929.34. Hubble E.P. and Humason M.L. Astrophysical Journal, 74, 43, 1931.35. Shapley H. and Ames A. Harvard Annals, 88, 43, 1932.36. Hubble E.P. Astrophysical Journal, 79, 8, 1934.37. Hubble E.P. Astrophysical Journal, 84, 517, 1936.38. Hubble E.P. and Tolman R.C. Astrophysical Journal, 82, 302, 1935.39. McVittie G.C. Monthly Notices, 97, 163, 1937.40. McVittie G.C. Cosmological theory. Ch. IV. Methuen Monographs,

London, 1937.41. McVittie G.C. Monthly Notices, 98, 384, 1938.42. McVittie G.C. Zeitschrift fur Astrophysik, 14, 274, 1937.43. McCrea W.H. Zeitschrift fur Astrophysik, 9, 290, 1935.44. Eddington A. S. Monthly Notices, 97, 156, 1937.45. Shapley H. Proc. Nat. Acad. Sci. USA, 24, 148, 1938.46. Shapley H. Proc. Nat. Acad. Sci. USA, 24, 527, 1938.47. Shapley H. Monthly Notices, 94, 791, 1934.48. Shapley H. Proc. Nat. Acad. Sci. USA, 21, 587, 1935.49. Shapley H. Proc. Nat. Acad. Sci. USA, 19, 389, 1933.50. Shapley H. Proc. Nat. Acad. Sci. USA, 24, 282, 1938.51. McCrea W.H. Zeitschrift fur Astrophysik, 9, 98, 1939.52. Mason W.R. Philosophical Magazine, 14, 386, 1932.53. Fesenkov V.G. Doklady Acad. Nauk USSR, new ser., 15, 125, 1937.54. Fesenkov V.G. Soviet Astron. Journal, 14, 413, 1937.55. Krat V.A. Proc. Phys.–Math. Soc., Kiev Univ., ser. 3, 1936–37.56. Eigenson M.S. Doklady Acad. Nauk USSR, new ser., 26, 759, 1940.57. Cartan E.-J. La geometrie des espaces de Riemann (The geometry of

Riemannian spaces). ONTI, Moscow–Leningrad, 1936.58. Lamb H. Hydrodynamics. 5th ed., Cambridge Univ. Press, Cam-

bridge, 1930.59. Lanczos K. Physikalische Zeitschrift, 23, 537, 1922.60. Fock V.A. Soviet Physics JETP–USSR, 9, 375, 1939.61. Eisenhart L.P. Trans. Amer. Math. Soc., 26, 205, 1924.62. Kochin N.E. Vector calculus and the foundations of tensor analysis.

6th ed., GONTI, Moscow–Leningrad, 1938.63. Cartan E.-J. Integral invariants. GITTL, Moscow–Leningrad, 1940.64. Landau L.D. and Lifshitz E.M. The classical theory of fields. GITTL,

Moscow–Leningrad, 1939.

Index

absolute rotation 195— chr.inv.-velocity 143, 195

accompanying frame of referen-ce 184

— local 183anisotropy

— degree of anisotropy 195— geometrical 195— of the curvature 199— of deformations 195, 199— mechanical 195

chr.inv.-curvature— invariant 114— of the 2nd rank 113— of the 4th rank 113

chr.inv.-differentiation 68— chr.inv.-operators 70

chr.inv.-divergence 92— physical divergence 173

chr.inv.-form of the equation ofgravitation 177, 180

chr.inv.-metric tensor 78chr.inv.-rotor 99chr.inv.-tensor of the rate of de-

formations 82chr.inv.-tensors 66chr.inv.-vector of velocity 81co-differentiation 68cosmological equations

— chr.inv.-form 192— of energy 49, 190— of gravitation 49, 190

cosmological principle 17co-tensors 66curvature of space 121

Einstein’s chr.inv.-tensor 96

force vector and the tensor 73Friedmann’s case 50

gravitational inertial force, itschr.inv.-vector 143

in-tensors 54

large scale 18locally-stationary system of ref-

erence 84

maximally orthogonal spatial sec-ctions 121

potentials 67— the method of variation

of potentials 68

Ricci’s chr.inv.-tensors 115Riemann-Christoffel tensor

— covariant 95— mixed 95

sample principle 34space of reference 64sub-tensors 64

theories of the expanding Univ-erse 17

theorem on space deformation 86total differentiation with respect

to time 132

Zelmanov’s theorem 66

The Classical Theory of Fields Revision Project (CTFRP)Collected Papers Treating of Corrections to the Book

“The Classical Theory of Fields” by L. Landau and E. Lifshitz

The “Course in Theoretical Physics” by L.Landau and E.Lifshitz has for decadesserved as a set of outstanding textbooks for students and reference for researchers.Many continue to learn their basic physics from this lucid and extensive expositionof physical theory and relevant mathematical methods.

The second volume of this series of texts, “The Classical Theory of Fields”,is a mainstay source for physicists learning or conducting research in GeneralRelativity∗. However, it has been realised over the years that “The Classical Theoryof Fields” contains a number of serious theoretical errors. The errors are in generalnot peculiar to this book alone, but are fundamental misconceptions that appearroutinely in all textbooks on General Relativity, without exception.

Save for the errors alluded to above, “The Classical Theory of Fields” remainsan authoritative and skilful exposition of Einstein’s theory of gravitation. To enhanceits already great standing in the scientific literature, the Editorial Board of Progressin Physics, an American journal on physics, proposes a series of papers dealing withcorrections of the now obsolete, although rather standard, erroneous argumentscontained in “The Classical Theory of Fields”. Any person interested in contributingto this project is invited to submit, for the consideration of the Editorial Board, apaper correcting one or more errors in the book. All papers will undergo review justas any research paper, and be published in Progress in Physics if accepted.

It is envisaged that accepted papers will also be collected together as a supple-mentary pamphlet to “The Classical Theory of Fields”, which will be made availablefree as a download from the Progress in Physics website. Each author’s contributionwill bear the author’s name, just like any research paper. All authors must agree tofree dissemination in this fashion as a condition of contribution.

Should the pamphlet, at any future time, be considered by the Publisher’sof the “Course in Theoretical Physics”, or any other publisher besides Progressin Physics, as a published supplement packaged with the “Course in TheoreticalPhysics”, all authors will be notified and can thereafter negotiate, if they wish,issues of royalties with the publisher directly. Progress in Physics will still reservethe right to provide the supplementary pamphlet free, from its website, irrespectiveof any other publication of the supplementary pamphlet by the publishers of the“Course in Theoretical Physics” or any other publisher. No author shall hold Prog-ress in Physics, its Editorial Board or its Servants and Agents liable for any royaltiesunder any circumstances, and all contributors will be required to sign a contractwith Progress in Physics to that effect, so that there will be no dispute as to termsand conditions. The Editorial Board of Progress in Physics shall reserve all rights asto inclusion or rejection of contributions.

Those interested in making a contribution should express that interest in anemail to the Editors of Progress in Physics who manage this project.

Dmitri Rabounski, Editor-in-ChiefStephen J. Crothers, Associate Editor

(the CTFRP organisers)

∗The first edition of “The Classical Theory of Fields” was completed in 1939,and published in Russian. Four revised editions of the book were later publishedin English in 1951, 1962, 1971, and 1975 (all translated by Prof. Morton Hamer-mesh, University of Minnesota). Reprints of the book are produced by Butterworth-Heinemann almost annually.

Abraham Zelmanov. Chronometric Invariants. Am. Res.Press, 2006, Edited by D.Rabounski and S. J.Crothers,232 pages. ISBN: 1-59973-011-1.

This book was written in 1944 by Abraham Zelmanov,a prominent scientist working in General Relativity andcosmology. Herein he constructs the theory of physicalobservable quantities in General Relativity (ChronometricInvariants), and applies it to determine all possiblecosmological models within the framework of Einstein’stheory — scenarios of evolution — which could betheoretically conceivable for a truly inhomogeneous andanisotropic Universe.

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